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UNIVERSITY OF CALIFORNIA,
IRVINE
Nonlinear Methods for High Resolution Spectral Analysis and Their Applications inNuclear Magnetic Resonance Experiments
DISSERTATION
submitted in partial satisfaction of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
in Chemistry
by
Jianhan Chen
Dissertation Committee:Professor Vladimir A. Mandelshtam, Chair
Professor Craig C. MartensProfessor Steven R. White
2002
c© 2002Jianhan Chen
The dissertation of Jianhan Chen
is approved and is acceptable in quality
and form for publication on microfilm:
Committee Chair
University of California, Irvine
2002
ii
To my grandparents, my parents,
and particularly, my wife, Xueying
iii
TABLE OF CONTENTS
ABSTRACT OF THE DISSERTATION 1
LIST OF FIGURES vi
LIST OF TABLES viii
ACKNOWLEDGEMENTS ix
CURRICULUM VITAE x
1 Introduction to Spectral Analysis 11.1 Definition and Significance . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Fourier Transform Spectral Analysis . . . . . . . . . . . . . . . . . . . 3
1.2.1 Fourier Transform of Discrete Signals . . . . . . . . . . . . . . 51.2.2 Data Manipulation . . . . . . . . . . . . . . . . . . . . . . . . 81.2.3 Multi-Dimensional DFT . . . . . . . . . . . . . . . . . . . . . 131.2.4 Limitations of FT . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3 Alternative Methods for Spectral Analysis . . . . . . . . . . . . . . . 221.3.1 Principle of High-Resolution Spectral Analysis . . . . . . . . . 221.3.2 Parametric Methods . . . . . . . . . . . . . . . . . . . . . . . 241.3.3 Nonparametric Methods . . . . . . . . . . . . . . . . . . . . . 351.3.4 Subspace Methods . . . . . . . . . . . . . . . . . . . . . . . . 38
1.4 Summary and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2 Theory of the Filter Diagonalization Method 422.1 Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.2 One-Dimensional FDM . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2.1 Nonlinear Fitting by Linear Algebra . . . . . . . . . . . . . . 462.2.2 Local Spectral Analysis: Fourier-Type Basis . . . . . . . . . . 512.2.3 FDM as a High-Resolution Spectral Estimator . . . . . . . . . 562.2.4 Multi-Scale Fourier-Type Basis . . . . . . . . . . . . . . . . . 672.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.3 Multi-Dimensional Extensions of FDM . . . . . . . . . . . . . . . . . 802.3.1 What Is a True Multi-Dimensional Method? . . . . . . . . . . 812.3.2 2D Harmonic Inversion Problem . . . . . . . . . . . . . . . . . 832.3.3 A “Naive” Version of 2D FDM . . . . . . . . . . . . . . . . . 842.3.4 Green’s Function Approach . . . . . . . . . . . . . . . . . . . 932.3.5 Averaging Approaches to Suppress the Artifacts . . . . . . . . 952.3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
2.4 Regularization of Multi-Dimensional FDM . . . . . . . . . . . . . . . 1022.4.1 Ill-conditioning of 2D FDM . . . . . . . . . . . . . . . . . . . 102
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2.4.2 Singular Value Decomposition . . . . . . . . . . . . . . . . . . 1052.4.3 FDM2K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1122.4.4 Snap-Shot FDM: Self-Regularization. . . . . . . . . . . . . . . 1162.4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
2.5 Regularized Resolvent Transform . . . . . . . . . . . . . . . . . . . . 1222.5.1 One-Dimensional RRT . . . . . . . . . . . . . . . . . . . . . . 1222.5.2 Multi-Dimensional RRT . . . . . . . . . . . . . . . . . . . . . 1312.5.3 Other Spectral Representations Using RRT . . . . . . . . . . 1352.5.4 Extended Fourier Transform . . . . . . . . . . . . . . . . . . . 1402.5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
2.6 Reliability and Sensitivity of FDM/RRT . . . . . . . . . . . . . . . . 1462.6.1 Reliability of 1D FDM and RRT . . . . . . . . . . . . . . . . 1462.6.2 Possible Ways for Checking Individual Peaks . . . . . . . . . . 1502.6.3 A Semi-Quantitative Way for Estimating the Sensitivity of FDM/RRT
Spectral Estimations . . . . . . . . . . . . . . . . . . . . . . . 1522.6.4 Reliability of 2D FDM . . . . . . . . . . . . . . . . . . . . . . 155
2.7 Summary and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 158
3 Application of FDM for NMR Spectral Analysis 1673.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1673.2 Application of FDM/RRT for 1D NMR . . . . . . . . . . . . . . . . . 171
3.2.1 Linear Phase Correction . . . . . . . . . . . . . . . . . . . . . 1713.3 Nontrivial Projections by FDM . . . . . . . . . . . . . . . . . . . . . 178
3.3.1 45o-Projections of J-Experiments. . . . . . . . . . . . . . . . . 1813.3.2 Singlet-TOCSY Spectra via FDM . . . . . . . . . . . . . . . . 183
3.4 Constant-Time NMR Signals Processed by FDM . . . . . . . . . . . . 1883.4.1 Special Properties of CT NMR Signals . . . . . . . . . . . . . 1893.4.2 A Doubling Scheme for Processing CT NMR Signals . . . . . 1923.4.3 Lineshape Modification . . . . . . . . . . . . . . . . . . . . . . 1963.4.4 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . 1973.4.5 Application to a CT-HSQC Experiment . . . . . . . . . . . . 2023.4.6 A Quadrupling Scheme for 3D NMR Experiments with Two
Constant-Time Dimensions . . . . . . . . . . . . . . . . . . . . 2143.5 Summary and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Bibliography 232
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LIST OF FIGURES
1.1 Two simple time signals and their Fourier transforms. . . . . . . . . . 41.2 Absorption and dispersive Lorentzian lineshapes. . . . . . . . . . . . 101.3 A general scheme for 2D NMR experiments. . . . . . . . . . . . . . . 141.4 Comparison of double-absorption, double-dispersion and phase-twisted
2D Lorentzian lineshapes. . . . . . . . . . . . . . . . . . . . . . . . . 171.5 Limitations of Fourier transform. . . . . . . . . . . . . . . . . . . . . 20
2.1 A FDM single window calculation. . . . . . . . . . . . . . . . . . . . 522.2 A multi-window implementation of FDM. . . . . . . . . . . . . . . . . 582.3 Spectral estimation via FDM: I(ω) or Iτ (ω). . . . . . . . . . . . . . . 612.4 The spectrum of triplets: Jacob’s Ladder. . . . . . . . . . . . . . . . . 632.5 A comparison of FT and FDM, applied to “Jacob’s Ladder”. . . . . . 642.6 A case where DFT beats FDM. . . . . . . . . . . . . . . . . . . . . . 652.7 An example of a multi-scale Fourier basis. . . . . . . . . . . . . . . . 692.8 Plot of imaginary part of complex frequency grid vs. effective FT
length for Multi-Scale Fourier basis with complex frequency grid. . . . 752.9 A demonstration of Multi-Scale FDM using an artificial noisy signal
with very strong background. . . . . . . . . . . . . . . . . . . . . . . 762.10 An FT-IR spectrum processed by Multi-Scale FDM. . . . . . . . . . . 782.11 Schematic illustration of a true 2D spectral analysis method. . . . . . 822.12 FDM is a true multi-dimensional spectral analysis method. . . . . . . 912.13 Spectra obtained using 2D FDM without regularization. . . . . . . . 962.14 Spectra obtained by FDM with pseudo-noise averaging. . . . . . . . . 992.15 Hard regularization of FDM via truncated SVD. . . . . . . . . . . . . 1092.16 Soft regularization of FDM via SVD pseudo-inverse of U0. . . . . . . 1112.17 Regularization of FDM via FDM2K. . . . . . . . . . . . . . . . . . . 1142.18 Comparison of Tikhonov Regularization and FDM2K. . . . . . . . . . 1162.19 Comparison of RRT and DFT, applied to “Jacob’s Ladder”. . . . . . 1272.20 Complex spectra of a single peak as a function of line width. . . . . . 1292.21 RRT spectral estimation with smoothing. . . . . . . . . . . . . . . . . 1302.22 Comparison of 2D RRT and DFT. . . . . . . . . . . . . . . . . . . . . 1342.23 Pseudo-2D RRT spectrum: I(ω + iΓ) on the complex plane. . . . . . 1372.24 RRT pseudo-absorptions spectrum. . . . . . . . . . . . . . . . . . . . 1392.25 Illustration of 1D XFT. . . . . . . . . . . . . . . . . . . . . . . . . . . 1422.26 FDM Spectrum is a perfect fit of FT spectrum. . . . . . . . . . . . . 1472.27 Comparison of the original signal and the signal reconstructed using
the spectral parameters obtained by FDM. . . . . . . . . . . . . . . . 1482.28 Sensitivity of FDM/RRT spectral estimations. . . . . . . . . . . . . . 1532.29 Sensitivity of FDM/RRT as a function of basis size and density. . . . 1542.30 Reliability of 2D FDM calculation. . . . . . . . . . . . . . . . . . . . 156
vi
3.1 Linear phase correction via DFT and FDM/RRT: applied to an NMRsignal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
3.2 Robust linear phase correction via FDM/RRT: applied to a model signal.1773.3 The singlet-HSQC spectrum of progesterone . . . . . . . . . . . . . . 1823.4 Pulse sequence of J-TOCSY-J and conventional TOCSY experiments. 1843.5 Demonstration of singlet-TOCSY using an experimental J-TOCSY-J
4D signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1863.6 Vertical 1D traces of conventional and singlet-TOCSY spectra. . . . . 1873.7 Sensitivity of CT experiments. . . . . . . . . . . . . . . . . . . . . . . 1903.8 Illustration of time symmetry between N- and P-type CT signals. . . 1933.9 An example of lineshape modification. . . . . . . . . . . . . . . . . . 1963.10 Demonstration of FDM doubling scheme using a high SNR model signal.1993.11 Demonstration of FDM doubling scheme using a low SNR model signal.2013.12 A modified CT-HSQC pulse sequence. . . . . . . . . . . . . . . . . . 2033.13 Demonstration of of the noise performance of FDM2K. . . . . . . . . 2053.14 Ultra-short constant-time spectra by FDM and FT. . . . . . . . . . . 2083.15 A ultra-short constant-time spectrum obtained by mirror-image LP. . 2103.16 Demonstration of FDM2K using a realistic protein, MTH1598. . . . . 2133.17 Real and imaginary parts of the first FID of a 3D HNCO signal. . . . 2163.18 2D (H,C) Projections of the 3D FT and 3D FDM spectra. . . . . . . 2183.19 2D (H,N) Projections of the 3D FT and 3D FDM spectra. . . . . . . 2193.20 2D (C,N) Projections of the 3D FT and 3D FDM spectra. . . . . . . 2203.21 Comparison of the finite FT (H,C) projection estimated from FDM
spectral parameters with the actual FT projection. . . . . . . . . . . 2233.22 Comparison of the finite (C,N) FT projection estimated from FDM
spectral parameters with the actual FT projection. . . . . . . . . . . 2243.23 Comparison of the long FT projection estimated by FDM using a short
signal with 6 × 8 (C,N) increments, with the actual FT projectionobtained from a long signal with 12 × 32 (C,N) increments. . . . . . . 225
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LIST OF TABLES
viii
ACKNOWLEDGEMENTS
First, I would like to thank my thesis advisors, Professors Vladimir A. Mandelsh-
tam and A. J. Shaka, for their support, encouragement, and inspiration during the
last four years. Particularly, I thank Professor Mandelshtam for his patience, and the
many long hours he spent talking with me about the research projects. I thank both
advisors for being great mentors. It has been a great honor to be their student.
Special thanks go to Professors V. Ara Apkarian, Craig C. Martens and Douglas
J. Tobias for for their enthusiastic support and encouragement. I am also grateful to
Professors P. Taborek, S. R. White, D. A. Brant, and R. B. Gerber for their support
during my study in UC-Irvine. Dr. Anna A. De Angelis and Dr. Daniel Nietlispach
are specially acknowledged for great collaborations and helpful discussions.
Also many thanks to current and former group members of Shaka’s and Man-
delshtam’s groups: Dr. Haitao Hu, Dr. Que N. Van, Dr. Anna A. De Angelis, Dr.
Joseph E. Curtis, Dr. Mari A. Smith. Geoff S. Armstrong, and Kristin E. Cano. I
am especially indebted to Dr. I-Feng William Kuo for making my experience with
computers enjoyable, to Dr. Joseph E. Curtis for providing an excellent template for
this manuscript, and to Geoff S. Armstrong for proofreading this manuscript.
Most importantly, I would like to thank my family for their love and support
during the whole journey of coming to US to pursue this degree. I am especially
indebted to my wife, Xueying Qin, for her love, sacrifice and companionship.
Finally, I would like to thank NSF, CA-BioStar and the Department of Chemistry
at UC-Irvine for financial support.
ix
CURRICULUM VITAE
JIANHAN CHEN
Education
2002 Ph.D. in Chemistry, University of California at IrvineConcentration: Chemical and Materials PhysicsResearch Advisors: V. A. Mandelshtam and A. J. ShakaDissertation title: “Nonlinear Methods for High-Resolution Spectral Anal-ysis and Their Applications in Nuclear Magnetic Resonance Experiments”.
2000 M.S. in Chemistry, University of California at IrvineResearch Advisors: V. A. Mandelshtam and A. J. ShakaDissertation title: “Filter Diagonalization Method as a High ResolutionSpectral Estimator”.
1998 B.Sc. in Chemistry, University of Science & Technology of ChinaResearch Advisors: Kaibin Tang and Yitai QianThesis title: “Synthesis and Characterization of New 123-Series Cuprates”.
Experience
1999-2002 Graduate Research Assistant, University of California at Irvine
1996-1998 Reasearch Assistant, University of Science & Technology of China
Awards
2002 The E. K. C. Lee Fellowship
2002 Regents’ Dissertation Fellowship
2001 American Chemical Society Travel Fellowship
2001 Physical Science Faculty Endowed Fellowship
2001 42nd ENC Student Travel Award
1998 Lian Xiang Scholarship
1993 Xu A’Quong Scholarship
Publications
1. J. Chen and V. A. Mandelshtam, ”Multi-Scale Filter Diagonalization Methodfor spectral analysis of noisy data with nonlocalized features”, J. Chem.Phys. 112, 4429 (2000).
2. J. Chen, V. A. Mandelshtam and A. J. Shaka, ”Regularization of the FilterDiagonalization Method: FDM2K”, J. Magn. Reson. 146, 368 (2000).
x
3. J. Chen, A. J. Shaka, V. A. Mandelshtam, ”RRT: The Regularized ResolventTransform for high resolution spectral estimation”, J. Magn. Reson. 147,129 (2000).
4. A. A. De Angelis, J. Chen, V. A. Mandelshtam and A. J. Shaka, ”A Methodto Obtain High Resolution 13C CT-HSQC Spectra of Proteins with a ShortConstant-Time Period”, J. Biomol. NMR (submitted).
5. J. Chen, A. A. De Angelis, V. A. Mandelshtam and A. J. Shaka, ”Progress onTwo-Dimensional Filter Diagonalization Method. An Efficient Doubling Schemefor Two-Dimensional Constant-Time NMR”, J. Mag. Reson. (submitted).
6. M. A. Smith, J. Chen, and A. J. Shaka, ”Small Molecule NOE DifferenceSpectroscopy for High-Resolution NMR”, J. Am. Chem. Soc. (submitted).
7. J. Chen, V. Mandelshtam and D. Neuhauser, ”Calculating excited vibrationallevels of polyatomic molecules by diffusion Monte Carlo”, J. Chem. Phys., inpreparation.
8. J. Chen, D. Nietlispach, V. A. Mandelshtam, and A. J. Shaka, “Ultra-highquality HNCO spectra with very short constant times”, in preparation.
Selected Presentations
1. ”A faster Filter Diagonalization Method with fully automated multi-windowing”,Poster Presentation, 41st Experimental Nuclear Magnetic ResonanceConference, April 2000.
2. ”RRT: The Regularized Resolvent Transform for high resolution spectralestimation”, Poster Presentation, 42nd Experimental Nuclear MagneticResonance Conference, March 2001.
3. ”The Regularized Resolvent Transform and its Applications”, Invited Talk,ACS Annual Meeting, August 2001.
4. ”The Filter Diagonalization Method for Spectral Analysis of NMR Data”,Invited Talk, Washington Area NMR Group, February 2002.
5. ”High Resolution Double Constant-Time 3D NMR Spectra by FilterDiagonalization Method”, Poster Presentation, 43nd Experimental NuclearMagnetic Resonance Conference, April 2002.
xi
ABSTRACT OF THE DISSERTATION
Nonlinear Methods for High Resolution Spectral Analysis and Their Applications in
Nuclear Magnetic Resonance Experiments
by
Jianhan Chen
Doctor of Philosophy in Chemistry
University of California at Irvine, 2002
Professor Vladimir A. Mandelshtam, Chair
Novel nonlinear methods for high resolution spectral analysis of time domain sig-
nals, namely, the Filter Diagonalization Method (FDM) and Regularized Resolvent
Transform (RRT), have been developed and applied to the signal processing prob-
lems in Nuclear Magnetic Resonance (NMR) experiments. Several important break-
throughs that significantly improve the efficiency and stability of FDM and RRT are
discussed. They include the multi-scale Fourier-type basis, regularization of multi-
dimensional FDM, and a doubling scheme for Constant-Time NMR signals. Applied
to several 2D and 3D protein NMR experiments, substantial resolution enhancement
has been demonstrated. Several qualitative and semi-quantitative ways of studying
the reliability and sensitivity of FDM and RRT are discussed. Various issues that are
specific to NMR spectral analysis are also addressed.
xii
Chapter 1
Introduction to Spectral Analysis
1.1 Definition and Significance
Spectral analysis is the process of obtaining spectral information, such as charac-
teristic frequencies, amplitudes and phases, from one or more time-varying signals.
Spectral analysis plays an important role in many fields of science and engineering.
Time domain signals are often acquired in modern experiments, including Nuclear
Magnetic Resonance (NMR), Ion Cyclotron Resonance (ICR) and others. Time sig-
nals also occur in numerical simulations of dynamic systems, such as in unimolecular
reactions [1], dissociation of vibrationally excited van der Waals complexes [2], and
many other chemical processes [3]. Spectral analysis is the key to revealing the infor-
mation encoded in these signals. The goal of spectral analysis is to extract maximum
information from the available time domain data, or, to minimize the required ex-
perimental or computational time for extracting the same amount of information.
The conventional Fourier Transform (FT) [4] has played an essential role in spectral
1
analysis and was crucial to the development of many modern spectroscopies includ-
ing NMR, optical and mass spectrometry [5]. FT is a linear transformation that is
stable, reliable and fast if implemented as the Fast Fourier Transform (FFT) [6, 7].
In addition, FT does not make any assumptions about the functional form of the
signal and can be applied to processing any type of signal. Accordingly, FT is a very
powerful method and has been widely used for spectral analysis in many fields.
However, FT also has some well-known limitations. The FT resolution is limited
by the FT time-frequency uncertainty principle. Also, truncation of the signal leads
to sinc-like oscillative artifacts in the baseline of the FT spectrum. Another limitation
of FT is that additional information about the signal, if exists, cannot be efficiently
incorporated and utilized. Furthermore, multi-dimensional FT is simply a series of
1D FT applied to each dimension independently. Thus, FT is essentially a 1D method
for spectral analysis and cannot utilize the important information that evolution in
all dimensions are actually correlated. Therefore, spectral analysis remains an active
research area [8]. A lot of effort has been devoted to developing various alternative
methods to FT spectral analysis, seeking to make use of some a priori knowledge and
to extract information more efficiently from the time signals.
The purpose of this chapter is to introduce the existing methods for spectral anal-
ysis, with mainly the application of NMR spectroscopy in mind. We will first discuss
the basic properties and major limitations of the conventional FT spectral analysis
in Section 1.2, then briefly review some existing alternative methods in Section 1.3.
Finally, some conclusions and remarks are given in Section 1.4.
2
1.2 Fourier Transform Spectral Analysis
A physical process can be either described by a time domain function, c(t), where
t is the time variable in seconds, or a frequency domain function, I(ω), where ω is the
angular frequency in radians per second. To be general, we consider both functions
to be complex. These two functions form a “Fourier pair” as they are connected by
the Fourier transform equations:
I(ω) =∫ ∞
−∞c(t) eiωt dt ,
c(t) =1
2π
∫ ∞
−∞I(ω) e−iωt dω . (1.1)
I(ω) is also called a spectrum, or a spectral representation of the time signal. The
transformation from c(t) to I(ω) is often called the (forward) Fourier transform, and
its inverse is often called the inverse Fourier transform.
From Eq. 1.1, it is obvious that Fourier transformation is a linear operation. There
is no information gained or lost by transforming from one representation to the other:
the two functions contain exactly the same information but in different formats. The
important difference is that features that are delocalized in one function become
localized in the other. Interestingly, this becomes very important when it comes to
interpreting the information encoded in these functions. In a typical time signal,
contributions from different constituents, or resonances, are highly delocalized and
strongly interfere with each other, making direct interpretation of the signal very
difficult. For example, Figure 1.1 plots two simple time signals: one with single
resonance, trace (a), and the other one with two resonances of equal intensities and
decay rates, trace (b). When there is only a single resonance, it is possible to estimate
3
0 10 20 300 5 10Time (second) Frequency (Hz)
(a)
(b)
J=1 Hz
2.0 sec
Figure 1.1: Two simple time signals and corresponding Fourier transform spectra(only real parts of the complex functions are shown). While contributions from dif-ferent components are delocalized and interfere with each other in the time domain,they are highly localized and separated according to the characteristic frequencies inthe frequency domain.
the frequency by counting the zero-crossing and the linewidth from the decay rate.
When there are two components with equal intensity present, it is possible to use the
“beating”, a consequence of interference, to estimate the separation of two frequencies.
But it becomes difficult to tell much about the characteristics simply by looking at the
time signal, other than some vague judegements, such as that one or more components
are narrow if the signal decays slowly. When more than two components with different
intensities and decaying rates are present, it becomes impossible to interpret the
time domain signals directly. On the contrary, features are highly localized and
4
separated according to the characteristic frequencies in the frequency domain, making
the interpretation of them much easier.
Now it is clear that spectral analysis, the process of obtaining a spectral rep-
resentation from the time domain signal, is a key step in making the information
interpretable, even though such a transformation does not magically increase the
information content by any means. FT is a natural choice for performing such a
transformation. However, as it will become clear later, it is not the only way, nor the
optimal one, for obtaining the spectrum in many cases. One of the main focuses of
this dissertation is to exploit alternative methods for extracting more (interpretable)
information from the available data set, when the signal satisfies certain known con-
straints.
1.2.1 Fourier Transform of Discrete Signals
In the most common situations, the time signal c(t) is sampled (i.e., its value is
recorded) on an equidistant time grid, tn = nτ = 0, τ, . . . , (N − 1)τ , resulting in a
discrete signal c(n) ≡ c(nτ). The interval τ is also called sampling rate, or dwell time
in NMR terminology. The fact that the signal is discrete instead of continuous has
profound implications on its Fourier transform.
Sampling Theorem and Aliasing
For any sampling rate τ , there is a special frequency ωc, called the Nyquist critical
frequency, given as,
ωc ≡π
τ(1.2)
5
The Nyquist critical frequency is important for two reasons. The first one is known
as the sampling theorem: “If a continuous function c(t), sampled at an interval τ ,
happens to be bandwidth limited to frequencies in magnitude smaller than ωc, i.e., if
I(ω) = 0 for all |ω| ≥ ωc, the the function c(t) is completely determined by its samples
c(n).” [9]. The explicit formula for c(t), given in terms of c(n), is,
c(t) = τ∞∑
n=−∞
c(n)exp[ωc(t − nτ)]
π(t − nτ). (1.3)
Therefore, even though generally speaking, the information content of a discrete signal
is infinitely smaller than that of a continuous signal, entire information content of the
bandwidth limited signal can be recorded in a discrete signal, sampled at at rate
τ−1 ≥ 2 ωmax. Fairly often, this is the case. For example, NMR signals are always
filtered and amplified in a finite frequency range before being sampled.
Another consequence of discrete sampling is a phenomenon called aliasing for sig-
nals that are not bandwidth limited to less than the Nyquist critical frequency: any
frequency component outside of the Nyquist frequency range, (−ωc, ωc), is aliased (or
falsely translated) into that range. In another word, frequencies that differ by mul-
tiples of 2 ωc are completely indistinguishable due to the discrete sampling. Aliasing
causes ambiguity in determining the true frequencies and should be avoided in most
circumstances. In some special cases, aliasing can be used on purpose to reduce the
spectral range and thus increase the digital resolution of the FT spectrum [10, 11].
6
Discrete Fourier Transform
Given a finite discrete time signal c(n), available from n = 0, 1, . . . , N − 1, we can
estimate its Fourier transform on the same numbers of discrete frequency points,
I(k) = I(ωk) =∫ ∞
−∞c(t) eiωkt dt ≈ τ
N−1∑
n=0
c(n) einωkτ , (1.4)
where ωk = 2πk/Nτ, k = −N/2, . . . , N/2. The summation of Eq. 1.4 is called the
Discrete Fourier Transform (DFT) of the N data points c(n). The inverse discrete
Fourier transform to recover c(n) from I(k) is:
c(n) =1
N
N−1∑
k=0
I(k) e−inωkτ . (1.5)
DFT is a linear transform of the time domain data, and is thus stable and reli-
able. However, direct calculation of DFT is an order N2 process. A more efficient
implementation is called the Fast Fourier Transform (FFT) [6, 7], which requires
only O(N lnN) operations. FFT requires N to be a power of two1. For data sets of
arbitrary sizes, the Fastest Fourier Transform in the West (FFTW) [12] can be used.
Resolution and Sensitivity
The two most important measurements of the quality of a spectrum are resolution
and sensitivity. Resolution refers to the ability to distinguish components of the signal
that are close in frequency. It can be intrinsically determined by the signal itself, or,
in turn, by the underlying physical system and experimental techniques used to probe
and detect the signal. It is also often affected by the numerical method used to obtain
1There are also FFT algorithms for data sets of length N that are not a power of two. But theystill require N being divisible by some other prime numbers [9].
7
the spectral representation. The obtainable resolution of a spectral analysis method
can be roughly measured by the Full-Width-at-Half-Height (FWHH) of the resulting
lineshape of an arbitrarily narrow line. The DFT resolution for a finite signal is
limited by the so called FT time-frequency uncertainty principle,
δω ∼ 1
Nτ. (1.6)
Thus, the resolution of DFT converges linearly with respect to the signal length. This
is one of the limitations of DFT. We will discuss its consequences in more details in
Section 1.2.4.
Sensitivity refers to the ability to distinguish signal from noise. A commonly
used measurement for sensitivity is the Signal-to-Noise Ratio (SNR) of the spectrum.
SNR can be defined as the ratio between the height of the highest signal peak to
the standard deviation of the noise. As pointed out by Hoch and Stern [8], while
SNR is often used to quantify the sensitivity, they are not the same. When nonlinear
methods like maximum entropy reconstruction or others are used to compute the
spectrum, SNR is not a good measurement of the sensitivity any more. In these
cases, sensitivity should be accessed explicitly, rather than inferred from the SNR of
the resulting spectrum.
1.2.2 Data Manipulation
Direct Fourier transform of the time signal is often unsatisfactory, mainly due
to many potential imperfections of the data. It is routine practice to “pre-process”
the data set before it is Fourier transformed. In this section, we will discuss several
8
common data manipulations, especially in NMR data processing.
Zero- and First-Order Phase Correction
In order to understand the concept of phase corrections, we need to first dis-
cuss two different lineshapes in NMR. Nuclear magnetic resonances in liquids can be
modeled as damped harmonic oscillators. Let’s consider a signal that has a single
component with frequency ω0 and decay constant γ0 > 0. It can be written as,
s(t) = exp[−iω0t − γ0t] = exp[−i(ω0 − iγ0)t] , (1.7)
for t ≥ 0; s(t) = 0 for t < 0. Such a signal is also called a damped sinusoid. Its Fourier
transform is then,
I(ω) =∫ ∞
0s(t) exp(iωt) dt =
1
i(ω − ω0) + γ0
=γ0
(ω − ω0)2 + γ20
+ i(ω − ω0)
(ω − ω0)2 + γ20
= A(ω) + iD(ω) , (1.8)
where A(ω) and D(ω) are called absorption and dispersion Lorentzian lineshapes,
illustrated in Figure 1.2. For large frequency offsets, the absorption Lorentzian line-
shape decays as 1/(ω − ω0)2, while the decay of dispersion Lorentzian lineshape is
proportional to 1/(ω − ω0). Accordingly, absorptive Lorentzian lineshape is more
localized and thus preferred for high-resolution NMR. In addition, the zero crossing
at the center of the dispersive Lorentzian lineshape is also unfavorable.
Experimental signals are not always as perfect as Eq. 1.7. The signal might not
have zero phase at t = 0 and it might not even start at t = 0. A more general model
for the signal is,
s(t) = exp[i(φ0 + ω0 t0)] exp[−i(ω0 − iγ0)t] , (1.9)
9
Re[I(ω)] Im[I(ω)]
Figure 1.2: Real and imaginary part of the Fourier transform of a damped sinu-soid with zero phase at t = 0: (a) absorption Lorentzian lineshape; (b) dispersionLorentzian linshape.
where φ0 is called the zero-order phase, and ω0 t0 the first-order phase. The first-order
phase is due to the delay t0 and also called the linear phase as it has linear dependence
on the frequency. According to the linear property of FT, the Fourier transform of
s(t) is given as,
I(ω) = exp[i(φ0 + ω0 t0)] [A(ω) + iD(ω)] . (1.10)
Note that the real part of I(ω) automatically yields absorption lineshapes only when
both zero- and first-order phase are zero, or more precisely, φ0 +ω0 t0 = 0. Otherwise,
the real part of the complex spectrum will contain unfavorable dispersion lineshapes,
requiring the zero- and first-order phase corrections. The first-order phase correction
10
is also called linear phase correction. The zero-order phase can be reliably removed
simply by multiplying I(ω) by exp(−iφ0). However, the first-order phase can only be
approximately corrected in FT. The reason is that exact first-order phase correction
requires knowledge of the frequencies of all components, which is not available in
FT spectral analysis. In practice, we simply multiply I(ω) by exp(−iω t0) to remove
the first order phase. Such a point-by-point correction will be exact only at the
center of each resonance. A large linear phase correction leads to severe phase roll
in the baseline. Such an example is given in Figure 3.1, trace (c). The linear phase
occurs very often in NMR due to many experimental limitations including limited
transmitter response time, stabilization delay after field gradient pulses and others.
One of the main goals of developing alternative methods to FT is actually to find a
more consistent way of correcting the linear phase.
Zero-Filling
The digital resolution of the DFT spectrum is given as δω = 1/Nτ and thus
limited by the number of data points in the time domain. We can append a sequence
of zeros to the available data set to arbitrarily increase the total number of data
points and thus improve the digital resolution of the resulting DFT spectrum. Such
a process is called zero filling. Zero-filling is equivalent to some interpolation in the
frequency domain. At the frequency points they have in common (n/Nτ), the DFT
spectra of the zero-filled and original data will agree.
NMR signals obeys the causality principle as c(t) = 0 for all t < 0. As a conse-
quence, the real and imaginary parts of the complex FT spectrum have a deterministic
11
relationship according to the Kramers-Kronig relations [13, 14]. However, as noted by
Bartholdi and Ernst, the Kramers-Kronig relations do not hold for discretely sampled
NMR signal unless the signal is extended by a factor of two by zero-filling, because
the periodicity of DFT makes the real and imaginary parts independent [15]. It was
proved that real improvement in the information content was obtained by zero-filling
the signal by a factor of two. Further zero-filling results in only cosmetic interpola-
tion between data points in the frequency domain and no additional information is
obtained.
Apodization
Apodization is the process of modifying the original data by multiplying it with a
apodization function before zero-filling and Fourier transform:
c′(n) = c(n) w(n) .
w(n) is also called windowing function or filter function. The original DFT can be
treated as a DFT with a rectangular windowing function, w(n) = 1. The purpose of
apodization is to improve the quality of certain aspect of the spectrum or to obtain
a desired lineshape. For example, a matched filter, an apodization function that
matches the envelope of the signal, is often used to enhance the sensitivity of the
resulting spectrum [16], with the side effect of broadening the lines. Or, an increasing
exponential apodization function can be used to improve the resolution, but on the
price of reducing the sensitivity. For truncated signal, it is often necessary to use
some apodization to reduce the amplitude of the signal smoothly to zero in order
12
to suppress the truncation artifacts2, which also inevitably broadens the lines. In
practice, one has to compromise among resolution, sensitivity, lineshape, amount of
truncation artifacts in finding an “optimal” apodization function. A comprehensive
overview of various commonly used apodization functions in NMR can be found in
Hoch and Stern’s seminar book on NMR data processing [8].
1.2.3 Multi-Dimensional DFT
It is the ability to perform multidimensional NMR experiments that makes NMR
such a powerful method as it is today. By introducing additional dimensions, we are
able to disperse the potentially heavily overlapping features in the one-dimensional
(1D) spectrum into the addition dimensions and thus have a much better separation.
We are also able to obtain important information about the chemical and/or dynamic
correlations between resonances. It is fair to say that without multidimensional NMR
experiments, many of the applications that fueled the explosive growth of NMR in
the last two decades would not be possible [8].
A D-dimensional complex signal, sampled discretely on a D-dimensional time grid,
can be written as,
c(~n) ≡ c(n1τ1, . . . , nDτD) , nl = 0, 1, . . . , Nl − 1 ,
where τl is the sampling rate along l-th dimension. In NMR, the last time variable
tD = nDτD is the acquisition time dimension by convention. The other time dimen-
sions are called indirect dimensions, because they can only be incremented indirectly
2See Section 1.2.4 for more details on truncation artifacts.
13
as varying delays during the experiment. Figure 1.3 shows a general scheme [11]
for recording a two-dimensional NMR signal. There are two time dimension, t1 and
t2. Only the acquisition time t2 is the real time. t1 is an indirect time dimension,
implemented as a varying delay. For each value of t1 = n1τ1, n1 = 0, 1, . . . , N1 − 1,
the whole experiment is repeated and a 1D signal, c(n1τ1, t2), is recorded. After a
total of N1 1D experiments, a 2D signal results. It is important to notice that the
Preparation evolution (t1) mixing acquisition (t2)
Figure 1.3: General scheme for 2D NMR spectroscopy. It includes four segments:preparation of the states, evolution in the indirect dimension (t1), mixing period, andacquisition time (t2). In the mixing period, a correlation is established between thetwo time dimensions. Only t2 is the real time.
acquisition time dimension is the only inexpensive dimension, sampled in real time at
a rate of one data point per ms. Each increment in the indirect dimension requires
repeating the 1D experiment over and is thus time consuming. By analogy, for a 3D
experiment, each increment in the second indirect dimension requires repeating the
whole 2D experiment over, and so on. The total experimental time of a D-dimensional
NMR experiment is thus proportional to the total number of increments in the in-
direct dimensions , N1 × N2 × . . . × ND−1. This is the reason why multidimensional
NMR signals are typically truncated in the indirect dimensions.
Surprisingly, despite all of its importance, processing of multidimensional time
signals by FT requires few new concepts beyond those involved in the 1D FT spectral
analysis. The complex D-dimensional DFT, defined as,
14
I(ω1, . . . , ωD) =D∏
l=1
τl
N1−1∑
n1=0
. . .ND−1∑
nD=0
ein1ω1τ1 . . . einDωDτD c(n1τ1, . . . , nDτD) , (1.11)
where ωl is a Nl-point frequency grid specified as ωlk = 2πk/Nlτl, k = −Nl/2, . . . , Nl/2,
can be simply treated as a series of 1D DFT, applied to each dimension successively:
c(t1, ..., tD) FT
c(t1, ..., ωD) FT
.... I(ω1, ..., ωD)FT
All the data manipulation such as phase correction, zero-filling and apodization can
be applied to each dimension independently.
Absorption Mode DFT Spectrum Using Multiple Data Sets
One of the few complications of multidimensional DFT is that the absorption
mode lineshape cannot be obtained simply by taking the real part of the appropriately
phased D-dimensional complex DFT spectrum. For the sake of simplicity, let’s only
consider the 2D case. Following Eq. 1.8, the 2D complex DFT spectrum computed
from a single 2D complex signal can be written as,
I(ω1, ω2) = (A1 + iD1)(A2 + iD2)
= (A1A2 − D1D2) + i(A1D2 + D1A2) (1.12)
with convenient notations Al = A(ωl), Dl = D(ωl), l = 1, 2. Therefore, the real
part of the 2D complex spectrum is a supposition of the double-absorptive, A1A2,
and double-dispersive, D1D2, 2D Lorentzian lineshapes. The resulting lineshape is
phase-twisted and extremely undesirable in high-resolution NMR spectroscopy, as the
15
dispersive contributions tail off very slowly and degrade the resolution significantly.
Figure 1.4 compares the (a) double-absorption, (b) double-dispersion, and (c) phase-
twisted 2D Lorentzian lineshapes.
In order to obtain the desired double-absorption lineshape, FT requires two in-
dependent 2D complex data sets, acquired from the same experiment. In NMR, the
evolution in the indirect dimension, t1, can modulate either the amplitude or the
phase of the signal recorded during t23. In the former case, two amplitude modulated
signals, cosine and sine modulated signals, are acquired, which, in the case of a single
resonance, can be written as,
Sc(t1, t2) = cos(ω10t1) exp(−γ10t1) exp[−i(ω20 − iγ20)t2] , (1.13)
Sc(t1, t2) = sin(ω10t1) exp(−γ10t1) exp[−i(ω20 − iγ20)t2] . (1.14)
In the later case, N- and P-type phase modulated signals are acquired,
SN(t1, t2) = exp(−iω10t1 − γ10t1) exp[−i(ω20 − iγ20)t2] , (1.15)
SP (t1, t2) = exp(iω10t1 − γ10t1) exp[−i(ω20 − iγ20)t2] . (1.16)
Note that the signs of the frequency in t1 are opposite for N- and P-signals. These
two data formats can be converted into each other simply by linear combinations.
For example,
Sc(t1, t2) = [SP (t1, t2) + SN(t1, t2)]/2 , (1.17)
Ss(t1, t2) = [SP (t1, t2) − SN(t1, t2)]/2i . (1.18)
Using linear combination of the DFT of both of the 2D data sets from the same
3In derivation of Eq. 1.12, a purely phase modulated signal was actually assumed.
16
(a): Α1Α2
(b): −D1D2
(c): Α1Α2 −D1D2
Figure 1.4: Comparison of (a) double-absorption, (b) double-dispersion, and (c)phase-twisted 2D Lorentzian lineshapes, for a 2D Lorentzian line at frequency(ω1, ω2) = (0, 0) Hz with linewidth being 1 Hz in both dimensions.
17
experiment, double absorption lineshape can be constructed. Here, a procedure for
obtaining an absorption mode spectrum from two amplitude modulated data sets,
assuming that both data sets are correctly phased, is outlined:
(i). Apply DFT in the t2 dimension for both data sets:
Sc(t1, ω2) = cos(ω10t1) exp(−γ10t1)(A2 + iD2) , (1.19)
Ss(t1, ω2) = sin(ω10t1) exp(−γ10t1)(A2 + iD2) . (1.20)
(ii). Construct an intermediate signal according to:
Snew(t1, ω2) = Re[Sc(t1, ω2)] − i Re[Ss(t1, ω2)] = A2 exp[−i(ω10 − iγ10)t1] . (1.21)
(iii). Apply DFT to the intermediate signal in the t1 dimension. The real part of the
result gives the desired absorption mode 2D spectrum:
A(ω1, ω2) = Re[I(ω1, ω2)] = Re[A2(A1 + iD1)] = A1A2 . (1.22)
First Point Correction
Another complication of multidimensional FT is also related to the problem of
obtaining an absorption mode spectrum. The approximation of DFT to the FT
integral given in Eq. 1.4 has a systematic error: the first point should be weighted
by 1/2 before being transformed. While this only causes a constant vertical offset
in 1D DFT, the amount of such an offset will vary for each row in the indirect
dimension, causing so called t1-ridges in the 2D FT spectrum. These ridges deteriorate
the resolution of an absorption mode spectrum significantly and should be removed,
simply by incorporating the first point corrections into the multi-dimensional DFT
summation,
18
I(ω1, . . . , ωD) =D∏
l=1
τl
N1−1∑
n1=0
. . .ND−1∑
nD=0
ein1ω1τ1 . . . einDωDτD (1.23)
× (1 − δn1,0
2) . . . (1 − δnD ,0
2) c(n1τ1, . . . , nDτD) .
1.2.4 Limitations of FT
The Fourier transform is fast and numerically stable, and can produce frequency-
domain spectra in a convenient representation. It is the method of choice for spectral
analysis in many applications including NMR. Nonetheless, FT also has some well
known limitations, leaving plenty of room for improvement.
Truncation Artifacts and FT Uncertainty Principle
One of the most severe limitations of FT spectral analysis occurs when the signal
is truncated before it fully decays to zero, or, more realistically, below noise level.
The effect of truncation can be modeled as a step function, r(t) = 1 for t ≤ Tmax, and
r(t) = 0 for t > Tmax. The acquired signal c(t) can be then represented as the product
of the signal (extending to t = ∞), s(t), and the step function r(t): c(t) = s(t)r(t).
According to the convolution theorem, the resulting finite FT spectrum is given by,
I(ω) =∫ ∞
−∞s(t)r(t)dt = S(ω) ∗ sinc(Tmaxω) , (1.24)
where the sinc function is defined as sinc(x) = sin(x)/x. Convolution of the sinc
function has two direct consequences on the finite time FT spectrum: first, it produces
severe oscillating truncation artifacts in the baseline. The artifacts are often called
sinc-wiggles or Gibbs oscillations. Second, it broadens all the peaks to be at least
19
0 10 20 300 5 10Time (second) Frequency (Hz)
(a)
(b)
δω ∼ 1 / T
Figure 1.5: Effects of truncation on the FT spectrum: (1) sinc-wiggles in the baseline;(2) all lines are broadened to a minimum width of δω ∼ 1/Tmax. Note that the doubletof (b) is now unresolved due to the truncation of the signal.
the width of center envelope of the sinc function, which is roughly the inverse of the
signal length. The latter is often called the FT time-frequency uncertainty principle.
Figure 1.5 illustrates these two truncation effects using the same time signals used in
Figure 1.1, but truncated to 1/8 of the original sizes.
The sinc-wiggles are very unfavorable in high-resolution NMR as they obscure the
weak peaks around a strong peak. Apodization can be used to suppress the truncation
artifacts, which will further broaden the lines. Note that the resolution of finite FT,
determined by the FT uncertainty principle, improves only linearly with respect to the
signal length. While it does not look too bad in 1D, this slow convergence property
20
becomes a severe limitation of multi-dimensional FT spectral analysis.
1D Spectral Analysis
Multidimensional Fourier transform is simply a series of 1D FT, applied to each
dimension sequentially. The FT uncertainty principle applies to each dimension in-
dependently. Accordingly, in order to obtain high resolution in all dimensions of the
FT spectrum, one has to acquire a signal that is long in all dimensions. However,
as we discussed in the beginning of Section 1.2.3, the indirect dimensions of a multi-
dimensional NMR experiment are expensive and thus always truncated, or severely
truncated in the case of three- and four-dimensional experiments, leading to poor
FT resolution in the corresponding spectral dimensions. This is one of the major
limitations of multidimensional FT-NMR.
The major limitation of 1D spectral analysis is that correlations among multiple
dimensions are not explicitly exploited. The fact that evolutions in all dimensions are
correlated is completely ignored in FT spectral analysis. A true multi-dimensional
method for spectral analysis, which can efficiently use this information, can have
substantial advantages over any 1D methods including FT.
Modelless Spectral Analysis
FT does not make any assumptions about the signal. On one hand, this is a big
advantage as FT can be used to analyze any type of signal. On the other hand, this
can be a big disadvantage when the signal does satisfy certain models. For example,
an NMR signal in liquids can be well modeled by a sum of finite number of damped
21
sinusoids. Unfortunately, this useful information can not utilized and thus totally
wasted in the FT spectral analysis.
1.3 Alternative Methods for Spectral Analysis
Due the limitations of FT discussed in previous section, extensive efforts have
been made to develop alternative methods. One common goal of all these methods
is to produce resolution beyond the FT uncertainty principle, especially in the case
of truncated signals. We thus call these methods high-resolution methods for spectral
analysis. In this section, we will first discuss the basis principle of high resolution
spectral analysis and then briefly review major high-resolutions methods that have
been applied to NMR spectral analysis.
1.3.1 Principle of High-Resolution Spectral Analysis
A high-resolution method for spectral analysis is a method that can obtain a spec-
tral resolution beyond FT time-frequency uncertainty principle. In order to obtain
high-resolution, additional information must be used.
Let’s consider a simple example where we know that the signal contains a single
Lorentzian line and thus satisfies following model:
c(t) = d0 exp(−iω0t) , (1.25)
where the complex amplitude d0 = |d0| exp(iφ0) and complex frequency ω0 = 2πf0 −
iγ0 are two unknown parameters. Note that we extend the notation ω0 be complex so
22
that both the line position and linewidth can be represented by a single parameter.
The magnitude of the complex amplitude, |d0|, gives the peak integral, and φ0 defines
the phase of the peak. As shown in Figure 1.1 (a) and 1.5 (a), FT is not able to use
this information and cannot provide a fully converged spectrum4 unless the signal
fully decays (Figure 1.1 (a)). Alternatively, armed with the model of Eq. 1.25, we
know that only two unknowns, d0 and ω0, need to be found. In turn, it requires only
two data points, so that we can set up the two equations, from which both parameters
can be solved accurately:
c(0) = d0
c(τ) = d0 exp(−i ω0τ)
=⇒
d0 = c(0)
ω0 = i ln [c(τ)/c(0)] /τ
By analogy, for the slightly more complicated case of two Lorentzian lines (Figure 1.1
(b) and 1.5 (b)), as few as four complex data points are required to solve for four
unknowns: two pairs of complex amplitudes and complex frequencies, providing a
fully converged spectrum.
We just demonstrated that it was possible to obtain resolution beyond the FT un-
certainty principle by using some additional information that was not utilized in FT.
However, in practice, how we model the signal, how the model is used to reformulate
the problem, and how the problem is solved numerically are all very important limit-
ing factors. Experimental signals like NMR signals are often both large and complex,
and contain an unknown number of peaks. In addition, experimental signals do not
satisfy the model perfectly due to the noise and other imperfections. As a result, one
can easily end up solving a large, ill-conditioned linear or nonlinear problem, which
4A fully converged spectrum should show a peak at the right position with the true linewidth.
23
is computationally expensive and numerically unstable. In the rest of this section,
we will briefly review existing high-resolution methods, with the main focus being on
their main advantages and remaining difficulties.
The existing high-resolution methods can be grouped into two main categories:
parametric methods and nonparametric methods, based on whether a parametric
representation, a list of complex characteristic frequencies and amplitudes, is involved.
FT is a nonparametric method, as the spectral representation is directly obtained
without computing an intermediate parametric representation of the signal.
1.3.2 Parametric Methods
Practical methods for estimating the spectral parameters can be traced all the
way back to 1795, when Baron de Prony published his method for fitting the data
to a sum of complex sinusoids [17]. The idea of using linear algebra for identifying
the parameters of multi-exponential decay of a time signal is still used in more recent
parametric methods including Linear Prediction (LP) [18, 19, 20, 8], the matrix pencil
method [21, 22, 23], and the Filter Diagonalization Method (FDM) [24, 25]. The first
two methods will be described below, and FDM will be introduced later in Chapter 2.
Other parametric methods rely on nonlinear optimization of complex score functions,
based on either probability theories such as in Bayesian and maximum likelihood
techniques [26, 27, 28, 29], or simple minimum variation principle like in Least Squares
(LSQ) fitting in the frequency domain [30] and the time domain [31].
24
All these methods model the NMR signal as a sum of Lorentzian lines:
c(n) =∑
k
dk e−inωkτ + ξ(n) =∑
k
dk unk + ξ(n) , (1.26)
where dk and ωk are complex amplitudes and complex frequencies, and ξ(n) is noise.
The line list ωk, dk is called a parametric representation of the signal c(n). Each
pair of (ωk, dk) represents a peak, or, a pole. This model has been proved to be a
good model for most NMR signals in liquids. It is the additional information that
all these methods utilize, in one way or another. However, how this model is used to
reformulate the problem and the numerical approach used to solve for the unknown
spectral parameters vary from one method to another, and determine the efficiency
and stability of each method.
Linear Prediction
Linear prediction is one of the most popular high-resolution alternatives for NMR
spectral analysis. Even though linear prediction is typically only use to extrapolate
the signal, LP is intrinsically a parametric method. It is based on the concept of
autoregressive modeling (AR) [17, 16]:
c(n) =K∑
k=1
ak c(n − k) + ε(n) , n = K, . . . , N − 1 , (1.27)
where the coefficients ak are called LP coefficients, K is the prediction order, and
ε(n) is the prediction error. It can be proved that the AR model is equivalent to
the Lorentzian model of Eq. 1.26 with K damped sinusoids. Once the prediction
coefficients are determined, the spectral representation can be directly computed, or
25
more commonly, the AR expression is used to extrapolate the signal, typically, by a
factor of 2, then the FT of the extended signal is computed.
There exist several LP algorithms for finding the prediction coefficients including
LP-SVD [32], LP-QRD [33] and LP-TLS [34]. Detailed descriptions can also be found
in Refs. [8] and [20]. The basic idea is to find a set of coefficients ak that minimizes
the prediction error by solving a linear least squares problem,
c(K − 1) c(K − 2) · · · c(0)
c(K) c(K − 1) · · · c(1)
......
. . ....
c(N − 2) c(N − 3) · · · c(N − K − 1)
a1
a2
...
aK
=
c(K)
c(K + 1)
...
c(N − 1)
+
ε(K)
ε(K + 1)
...
ε(N − 1)
,
or in a compact form,
H x = b + ε . (1.28)
The complex frequencies ωk of all poles can be then computed by rooting the
characteristic polynomial, defined as,
P (u) =K∑
k=1
ak uK−k , (1.29)
where the roots uk = exp(−iωkτ). The complex amplitudes dk are determined by
solving another linear least squares problem based on Eq. 1.26:
1 1 · · · 1
u1 u2 · · · uK
......
. . ....
uN−11 uN−1
2 · · · uN−1K
d1
d2
...
dK
=
c(0)
c(1)
...
c(N − 1)
+
ξ(0)
ξ(1)
...
ξ(N − 1)
. (1.30)
26
Eqs. 1.29 and 1.30 might seem necessary only when LP is used as a parametric
estimator, while for LP extrapolation, only Eq. 1.28 needs to be solved. This is
not true. Some roots of the characteristic polynomial P (u) might fall outside of
the unit circle on the complex plane, i.e., |uk| > 1, which correspond to increasing
exponentials. In this case, direct use of the prediction coefficients to extrapolate the
signal is unstable and can result in numerical overflow. In practice, all the roots of
P (u) are first found, then some procedures have to be used to eliminate unwanted
roots. For example, those uk lying outside of the unit circle are either discarded or
replaced by their images reflected about the circle: uk → [u−1k ]∗. Finally, a new set of
prediction coefficients are computed by reconstructing the characteristic polynomial
using the “corrected” roots:
P ′(u) = (u − u1)(u − u2) . . . (u − uK) . (1.31)
The main advantage of LP is that the highly nonlinear problem of fitting the sig-
nal to the summation of damped sinusoids is solved by a linear fitting of the signal to
the AR model, and thus avoids the notoriously finicky nonlinear fitting procedures.
However, there are also several severe limitations:
(i). Filter order is a complete part of the AR model, and finding its correct value is
a crucial issue. Even though extensive efforts have been made in developing methods
for order selection [35, 36, 37, 38], no universal solution exits [20].
(ii). LP attempts to fit the whole 1D signal in one shot. The size of the least
squares problem is determined by the signal size and the filter order. Thus LP can
be prohibitively expensive for very large data sets. In addition, rooting high degree
27
polynomial requires significantly higher algorithm complexity necessary to avoid nu-
merical instability, round-off errors and overflow.
(iii) When parametric LP is used, it requires a second least squares calculation for
computing the complex amplitudes. Any errors in the rooting step might be ampli-
fied, making the parametric LP very unreliable.
(iv) In principle, it is possible to use LP for complete analysis of multidimensional
data sets. However, it is not in common practice due to the expensive computational
cost and numerical instabilities. Instead, hybrid LP and FT spectral analysis is typi-
cally used. For example, in 2D, LP is used to extrapolate each t1 trace independently
after FT is applied to the t2 dimension. Such a hybrid method is essentially a 1D
method and does not have the advantages of true multidimensional data processing.
(v) Even though the prediction coefficients are optimized by various tricks, the noisy
tail part of the signal is always used for extrapolation. Thus the extrapolation is
doomed to be noisy and unreliable.
The Matrix Pencil Method
The matrix pencil method was developed independently by Hua and Sarkar [21, 22]
and by Kailath and coworkers [39]. Instead of rooting a polynomial to obtain the
complex frequencies, they are computed directly from the data matrices by solving a
generalized eigenvalue problem. Its formulation exploits of the property of the matrix
pencil constructed from the underlying exponentially damped sinusoids (Eq. 1.26).
The matrix pencil refers to linear combinations of two matrices defined on a common
domain (i.e., same linear space). For example, given two matrices L and R, the set
28
of all matrices of form L− λR is said to be a matrix pencil.
Given a discrete time signal c(n) with N data points that satisfies Eq. 1.26, and
ignoring the noise term ξ(n), the (N − L) × L data matrices X0 and X1, which are
defined as:
Xl =
c(L − 1 + l) c(L − 2 + l) · · · c(l)
c(L + l) c(L − 1 + l) · · · c(1 + l)
......
. . ....
c(N − 2 + l) c(N − 3 + l) · · · c(N − L − 1 + l)
, (1.32)
l = 0, 1, with L being the pencil parameter, can can be decomposed as [21],
X0 = ZLBZR , X1 = ZLBZZR , (1.33)
where
ZL =
1 1 · · · 1
u1 u2 · · · uK
......
. . ....
uN−L−11 uN−L−1
2 · · · uN−L−1K
, ZR =
uL−11 uL−2
1 · · · 1
uL−12 uL−2
2 · · · 1
......
. . ....
uL−1K uL−2
K · · · 1
, (1.34)
and B = diagb1, b2, . . . , bK, Z = diagu1, u2, . . . , uK. If K ≤ L ≤ N − K, each of
uk : k = 1, 2, . . . , K is the rank reducing number of the matrix pencil X1 − uX0:
X1 − uX0 = ZLB
u1 − u 0 · · · 0
0 u2 − u · · · 0
......
. . ....
0 0 · · · uK − u
ZR . (1.35)
29
Therefore, by definition, uk are the K nonzero generalized eigenvalues of the matrix
pair (X1,X0),
X1 qk = ukX0 qi , (1.36)
where qk is the eigenvector associated with the eigenvalue uk. Note that Eq. 1.36 can
be solved by first multiplying both sides by a pseudo-inverse of X0, X+0 , and then
using standard eigenvalue solvers to find the K nonzero eigenvalues of L × L matrix
product X+0 X1:
X+0 X1 qk = uk qk . (1.37)
Finally, the complex amplitudes dk can be computed by solving the linear least squares
problem of Eq. 1.30. Note that spurious poles with |uk| > 1 can lead to numerical
instabilities in solving Eq. 1.30, and should be eliminated in similar ways to those
employed in linear prediction.
While sharing the advantages of being a linear algebraic method with LP, the
matrix pencil method has several advantages over LP. First, the pencil parameter L
will affect the results, in a similar way that LP is affected by the prediction order.
However, the performance of matrix pencil method is less sensitive to the choice of
L [21]. It was empirically found that the optimal value for L ranged from L = N/3
for noisy signals to L = N/2 for signals with high SNR. Second, in the matrix pencil
method, all the poles are computed directly from the data matrices in one step,
while LP requires first solving a linear least squares problem and then rooting the
resulting polynomial. Accordingly, the matrix pencil method is more efficient and
more accurate. It was also found that the matrix pencil method was less sensitive to
30
noise than the polynomial method [21, 23].
Nevertheless, the matrix pencil method still has some severe limitations, most of
which are shared by linear prediction. They are listed below:
(i). Estimating the pencil parameter L is still a problem, even though the results are
less sensitive to its value than in linear prediction.
(ii). The matrix pencil method still attempts to fit the whole signal in a single shot.
Thus for large data sets, it is computationally very expensive and has problems with
numerical instabilities, round-off error and overflow.
(iii). The amplitudes dk are computed by solving a separate linear least squares prob-
lem. Errors that are introduced in the diagonalization step might lead to disastrous
results in dk, making its application to analyzing NMR signals problematic5.
(iv). There are significant difficulties in generalizing the matrix pencil method to pro-
cessing multidimensional signals. Specifically, for example, in 2D, there are problems
in identifying the frequency pairs [22].
The Nonlinear Least Squares Analysis
Several least squares analysis of the DFT NMR spectra have appeared in the liter-
ature (see ref. [30] and references therein). Some of them are based on fitting the DFT
spectrum to the theoretical lineshape, Eq. 1.8, obtained from evaluating the infinite
time FT integral. However, as pointed out by Abildgaard and coworkers [40, 30],
there are several severe discrepancies between the DFT spectrum and the theoreti-
cal spectrum, which in many cases hamper these analyses. Instead, a more realistic
5The matrix pencil method was initially introduced to process acoustic and radar imaging signals,where only the complex frequencies were of general interest.
31
lineshape that corresponds to finite DFT of damped sinusoids should be used:
Iτ (ω) = τN−1∑
n=0
c(n) einωτ = τ∑
k
dk1 − (uk/u)N
1 − uk/u, (1.38)
where u ≡ exp(−iωτ). The object of least squares fitting in the frequency domain is
to minimize the variance between the estimated spectrum of Eq. 1.38 and the actual
DFT. Note that no zero-filling or apodization should be used in obtaining the actual
DFT spectrum. Least squares analysis can be also directly carried out in the time
domain by simply minimizing following loss function [31],
χ2(f) = χ2(K, ωk, dk : k = 1, 2, . . . , K) =N−1∑
n=0
∣
∣
∣
∣
∣
c(n) −K∑
k=1
dk e−inωkτ
∣
∣
∣
∣
∣
2
. (1.39)
It was claimed that the complete least squares analysis was the optimum spectral
estimator and maximum spectral information could be extracted [40]. While the
validity of such a statement is still under debate, there are other serious drawbacks
of the LSQ estimator. Firstly, LSQ is a nonlinear optimization method. Both an
accurate estimate of the number of pole and a good set of initial values for the spectral
parameters are required and essential for convergence. In the case of complicated
spectra with heavily overlapped features, such a requirement can not be fulfilled.
In addition, LSQ also attempts to fit the whole spectrum together in a single shot
and is thus computationally very expensive for large data sets. Finally, in principle,
LSQ can be applied to analyze multidimensional signals. However, due to the large
size and high complexity of multidimensional spectra, general application of LSQ to
multidimensional spectral analysis is totally infeasible.
32
Bayesian and Maximum Likelihood Methods
Bayesian analysis is a powerful statistical technique for estimating the degree to
which a hypothesis is confirmed by experimental data. It was introduced to NMR
spectral analysis in 1990 by Bretthorst [26, 27, 28] and Sibisi [41]. It has also been
applied to processing multidimensional NMR signals [29]. Only the basic principles
are described here. The details are quite complicated and can be found in the original
references or Hoch and Stern’s book on NMR data processing [8].
In essence, it is based on probability theory: given the data D and a particular
model M , according to Bayes’s rule, we have,
P (D)P (M |D) = P (D, M) = P (M)P (D|M) (1.40)
where P (D) and P (M) are the individual probabilities, P (D|M) and P (M |D) are
the conditional probabilities, and P (D, M) is the joint probability6. Accordingly, we
can calculate the conditional probability of the model for a given data set as
P (M |D) =P (D|M)P (M)
P (D)∼ P (D|M)P (M) . (1.41)
P (M |D) is also called the posterior of the model; P (D|M) is also called the likelihood
of the data (i.e., for a given model M , how likely the experimental data would be
obtained); and P (M) is also called the prior of the model (i.e., empirical or theoretical
estimation of the probability of the model M being true). P (D) is independent of
the model, and thus can be dropped out. The prior is only part the human a priori
knowledge can bias the results. It can be set to a flat function (such as a infinitely
6The conditional probability P (A|B) is the probability of A being true given that B is true. Thejoint probability P (A, B) is the probability of both A and B being true.
33
broad Gaussian function) so that no model is preferred. In this case, P (M) term can
be also dropped in the analysis.
Assuming that 1D NMR signals can be modeled as a sum of damped sinusoids7
of Eq. 1.26, the parameters of the model are then the number of peaks, K, and 2K
complex frequencies and complex amplitudes, ωk, dk. Thus,
P (M |D) ∼ P (D|M)P (M) = P (D|K, ωk, dk : k = 1, 2, . . . , K)P (M) . (1.42)
The calculation of the likelihood P (D|M) can be quite complicated. Details can
found in the original references. In order to find the best (most likely) model, i.e,
the number of poles and corresponding spectral parameters, for a given experimental
data, one needs to maximize the posterior, or equivalently, the likelihood in the case
of a flat prior function, by solving a high-dimensional optimization problem.
Despite the neat probability theory behind them, Bayesian and maximum like-
lihood techniques are hampered by the resulting nonlinear optimization problem.
Similar to LSQ method, finding a reasonable solution requires good initial set of
values for all the parameters, which is not available for complex signals. In addi-
tion, constructing a good likelihood funtion, P (D|M), is essential and can be tricky.
Finally, even though, in principle, these techniques can be applied to analyze any di-
mensional signals, the fast growing computational cost and resulting high dimensional
optimization problem make them practically infeasible.
7This assumption is equivalent to use a prior function P (M) = 1 for all Lorentzian models andP (M) = 0 for others.
34
1.3.3 Nonparametric Methods
The parametric approaches for high-resolution spectral analysis suffer from a com-
mon drawback: it is essential for the time signal to sufficiently satisfy the model em-
ployed. In the contrary, nonparametric methods like FT do not make any assumption
about the signal’s functional form, and are thus insensitive to the noise and lineshape
distortions of the signal. While there exist other nonparametric methods [42, 43, 44],
the most popular one in NMR spectroscopy is Maximum Entropy Reconstruction
(MaxEnt).
Maximum Entropy Reconstruction
The maximum entropy reconstruction for NMR spectral analysis was first intro-
duced by Sibisi in 1983 [45]. It can provide high resolution spectral estimation from
truncated data sets, and can evem be used to analyze signals that are nonuniformly
sampled. MaxEnt has generated considerable interest, and also some controversy on
the claims of simultaneous noise suppression and resolution enhancement. The de-
tails of the method have been described extensively [46, 47, 19, 8]. Here we merely
summarize the principles.
MaxEnt reconstructs the frequency domain spectrum, f(ω), directly by solving
an optimization problem similar to those in nonlinear least squares methods, with an
additional constraint of maximizing some entropy function, S(f). It is claimed that
the minimum variance principle along (such as Eq. 1.39) does not provide enough
information on the possible spectral reconstructions to be of much use. There exists
35
an infinite number of spectral reconstructions that satisfy the statistic constraint,
χ2(f) <N−1∑
n=0
|ξ(n)|2 = C0 ,
where ξ(n) is the noise in the data, and χ2(f) is defined as,
χ2(f) =N−1∑
n=0
∣
∣
∣[iDFT(f)]n − c(n)∣
∣
∣
2, (1.43)
where iDFT(f) is the “mock” time signal computed by inverse DFT of the spectrum
f(ω). If we are to improve over DFT, we have to include some additional criterion for
choosing an optimal reconstruction, such as the maximum entropy principle. What
maximum entropy principle implies is that a reasonable reconstruction should not
add any new information beyond what is contained in the experimental data [46, 8].
The entropy the spectrum f can be expressed in a general form,
S(f) = −N−1∑
n=1
R(fn) , (1.44)
where R(fn) is the negative of the contribution of each spectral point to the overall
entropy. The functional form of R(fn) resembles |fn| log(fn). Their actual forms can
be very complicated [47, 19, 8], but irrelevant to the present discussions.
The constrained optimization problem can be converted to an unconstrained one
of maximizing the following objective function,
Q(f) = S(f) − λ χ2(f) , (1.45)
where λ is a Lagrange multiplier that determines the relative weight of the entropy
and χ2-statistic constraint. The maximization problem of Eq. 1.45 does not have an
analytical solution and has to be solved numerically. Many algorithms have been
36
proposed [48, 8]. They typically rely on iteratively improving a trial guess of the
spectrum and simultaneously adjusting certain parameters such as λ to arrive at a
final solution.
There are certain attractions to MaxEnt. First, it does not make any assumption
about the signal to be recovered, while being flexible enough to incorporate some prior
information into reconstruction when such kind of knowledge is available. Second, a
good initial spectrum is not required for convergence, nor an accurate estimation of
the number of poles present in the signal. Third, it can be proved that there exists
a unique, global solution. Fourth, data sets that are not uniformly sampled can be
used, providing considerable flexibility to balance among sensitivity, resolution and
experimental time. However, the claim that MaxEnt can produce simultaneous noise
suppression and resolution enhancement [46] was proved to be largely unsubstantiated
due to the confusions between the sensitivity and SNR of the spectrum and between
the resolution enhancement and linewidth reduction [49, 8].
Nevertheless, there are also some severe limitations of MaxEnt:
(i). The efficiency and accuracy of MaxEnt hinge on the techniques used to solve the
high dimensional optimization problem. While theoretically a unique solution exists,
in practice, it relies on adjusting parameters such as the noise power C0, and there is
little guarantee that such a solution could be actually found.
(ii). MaxEnt is computationally very expensive, attempting to reconstruct the whole
signal in a single shot. Similar to all the previously described high-resolution meth-
ods, it is infeasible to use MaxEnt to analyze very large data sets.
(iii). As a result of its expensive computational cost, MaxEnt is typically only applied
37
in a trace-by-trace fashion for reconstruction of multidimensional spectra, losing the
advantages of true multi-dimensional spectral analysis.
(iv). Nonlinearity is intrinsic to MaxEnt, making it hard to use MaxEnt for quan-
titative purpose. Some techniques such as in situ calibration [48] were proposed to
overcome this limitation but only with limited efficiency.
(v). Even though additional information can be incorporated in MaxEnt, it is not used
in the optimal way. For example, when the signal sufficiently satisfies the Lorentzian
model, more aggressive methods like LP or the matrix pencil method can use this
information more efficiently and provide larger resolution enhancements.
1.3.4 Subspace Methods
Another common limitation of all the high-resolution alternatives we just dis-
cussed is the expensive computational cost. They all attempt to analyze the whole
signal in a single shot, i.e., in the full space, and can be prohibitively expensive and
very unstable for large data sets. In this section, we will discuss an example of sub-
space methods, LP-Zoom, that can avoid this limitation. Even though due to other
remaining difficulties, it is not actually successful enough for general spectral analysis
in NMR, the idea of converting a large problem into small problems in the subspace is
very interesting and one of the key reasons for the success of the filter diagonalization
method.
38
LP-Zoom
LP-Zoom [50] is based on LP theory but revised to analyze the local spectral
structures in contrast the global structure, and thus avoid the demands for large
computational time and memory of ordinary LP methods. The idea is as simple as
applying the z-transform to both sides of the AR model of Eq. 1.27,
g0(z) =K ′
∑
k=1
akgk(z) , (1.46)
where
gk(z) =N−1∑
n=K ′
c(n − k)zn , (1.47)
with z = exp(iωτ), and K ′ is a new prediction order. The essence of LP-Zoom is
that Eq. 1.46 allows the selective evaluation in a certain (small) frequency range of
interest. Because the zoomed area might contain only a few peaks, even if the whole
spectrum is very complicated, the required prediction order, K ′, is much smaller than
that of conventional LP. Selectively analyzing areas that contain peaks rather than
fitting the whole spectrum can significantly reduce the computational time and can
also yield a more accurate result.
However, other major difficulties of LP remain unsolved: 1). one still needs
to estimate the order of prediction filter, which is further complicated by the local
spectral analysis as an “effective rank” could be even more difficult to define. 2). the
amplitudes are still computed by solving another least square problem after all the
roots are solved. The errors in the rooting step could be disastrous for the second
step and are problematic to correct. 3). True multidimensional extensions still have
severe difficulties.
39
1.4 Summary and Remarks
Spectral analysis is an important step in extracting the spectral information from
the time signal. The conventional Fourier transform spectral analysis is a linear
method that is stable, reliable and fast if implemented as fast Fourier transform.
However, FT also has some well known limitations such as the FT time-frequency
uncertainty principle and sinc-like truncation artifacts. Moreover, FT is essentially a
one-dimensional spectral analysis method that is not able to use the important infor-
mation that evolution in all dimensions is correlated. Finally, additional information
about the signal, if exists, can not be efficiently incorporated in FT.
Extensive efforts have been made to develop high-resolution alternatives to the
FT spectral analysis. By utilizing some additional information about the signal, it is
possible to obtain resolutions beyond the FT uncertainty principle. However, due to
the large size and high complexity of NMR signals, none of these methods have been
completely successful. They all suffer from one or more of the following limitations:
(i). Rely on adjusting parameters to obtain satisfactory results.
(ii). Computationally very expensive and numerically unstable.
(iii). Rely on nonlinear optimization of some complex score functions with hun-
dreds of degrees of freedom.
(iv). Significant difficulties or limitations in their multidimensional extensions.
The Filter Diagonalization Method (FDM) is a high-resolution method that was
introduced to NMR spectral analysis only very recently. It seems to be free of all the
difficulties mentioned above, and shows the promise to be a final working method.
40
In the next two chapters, we will describe the theory of FDM and its application to
NMR spectral analysis in details. Here, a preview of FDM is provided:
(i). FDM is a parametric method based on the Lorentzian model of Eq. 1.26.
(ii). FDM solves the highly nonlinear fitting problem by converting it into pure
linear algebraic problems of diagonalizing some small data matrices in the frequency
subspace. It is thus intrinsically both computationally efficient and numerically sta-
ble.
(iii). In FDM, both the complex frequencies and complex amplitudes are obtained
simultaneously from the same eigenvalue problem.
(iv). Multi-dimensional extensions of FDM exist and have been successfully ap-
plied to analyze realistic NMR experimental signals. The whole multidimensional
data set is used by FDM to characterize the intrinsically multidimensional features.
Astonishing resolution enhancements have been demonstrated.
41
Chapter 2
Theory of the Filter
Diagonalization Method
2.1 Historical Overview
The filter diagonalization was first introduced by Neuhauser in 1990 to extract
eigenvalues and eigenstates of a potentially large system in any energy range of in-
terest [51]. In essence, a Fourier filter is applied to remove the correlation between
distant eigenstates, then, by diagonalizing the Hamiltonian matrix in the subspace,
the eigenstates in a small energy range can be computed very accurately. There are
two advantages of filter diagonalization: first, by filtering, the size of the Hamilto-
nian matrix is reduced; second, by diagonalization, it is possible to obtain accuracy
beyond the Heisenberg uncertainty principle, δE ∼ h/t, where t is the length of the
time propagation. One of the most successful early applications of filter diagonal-
ization was the accurate calculation of quantum scattering resonance of the HO2
42
radical [52], which is still very difficult to be reproduced by other methods. An
important breakthrough was made in 1995 by Wall and Neuhauser [24], where the
generation of a quantum time correlation function and its spectral analysis were split
into two independent steps. It was discovered that as long as the time correlation
function was available, the Hamiltonian matrix could be computed and diagonalized
to provide the eigenstates. In this new formulation, FDM became suitable for spec-
tral analysis of a general time signal, simply by ignoring the signal generation step.
However, this perspective remained only an interesting observation until its efficiency
was significantly improved by Mandelshtam and Taylor [25, 53]. The new version,
named the Filter Diagonalization Method (FDM), used a rectangular filter that was
computationally more efficient, and was reformulated for the conventional problem of
analyzing discrete time signals defined on an equidistant time grid. It was shown to
be an optimal method for fitting the signal to the summation of Lorentzian lines in
several points of view including convergence, computational efficiency and numerical
stability. FDM was then extensively exploited and applied to many problems in both
theoretical and experimental physics and chemistry, where the harmonic inversion
was essential [54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64]. In particular, FDM started
to be applied to processing Nuclear Magnetic Resonance (NMR) signals [65, 66, 67],
which turned out to be one of the most interesting applications of FDM.
From the spectral analysis point of view, 1D NMR does not really have limitations
as very long signals are typically available and therefore converged FT spectra1 can
be obtained. However, multi-dimensional NMR signals are often truncated in the in-
1A converged spectrum should show peaks at right frequencies with true linewidths.
43
direct dimensions due to many practical limitations, leading to poor FT resolution in
corresponding frequency dimensions. Therefore there is plenty of room for resolution
enhancement. Interestingly, the true power of FDM actually lies in its multidimen-
sional extensions. FDM is a true multidimensional method that is able to process the
whole multi-dimensional data set to characterize the intrinsically multi-dimensional
features. The obtainable resolution in all dimensions is determined together by the
total information content of the signal, instead of by the signal sizes along individual
dimensions. In another word, it is possible for FDM to use the information encoded in
the long dimensions to enhance resolution in the short dimensions. Early implemen-
tations of multi-dimensional FDM [68, 69, 70] require simultaneous diagonalization
of multiple data matrices, which is very difficult, if not impossible, for noisy signals
with degenerate resonances. A major breakthrough was made by Mandelshtam and
coworkers [71], when a Green’s function approach was introduced to compute mul-
tidimensional spectral estimations directly using the eigenvalues and eigenvectors of
solving several independent 1D generalized eigenvalue problems. The difficult prob-
lem of finding a unique set of eigenvectors to diagonalize multiple data matrices is
thus avoided. However, applied to noisy experimental signals, the spectra computed
from a single FDM calculation are often be contaminated by many artifacts including
spurious poles and distortions of genuine features, which are very sensitive to any
change in the input data and FDM processing parameters. The source of the arti-
facts was not fully understood at first, but the sensitivity was exploited by various
averaging methods to suppress the artifacts [72, 71, 73]. An obvious drawback of
these averaging methods was that they were very time consuming and the conver-
44
gence was slow. It was later realized that the averaging was actually an inefficient way
of regularization for removing the ill-conditioning of FDM. More efficient methods of
regularization were exploited, among which FDM2K [74], proposed in year 2000 and
thus named FDM2K to honor the new millennium, was particular efficient and ef-
fective. With regularization implemented, stable high-resolution spectral estimations
can be computed using single FDM calculations, provided that the signal sufficiently
satisfies the Lorentzian model and is sufficiently long. FDM2K has since been ap-
plied to several Constant-Time (CT) NMR experiments [75, 76] and demonstrated
significant resolution enhancements. There are some special properties associated
with processing CT-NMR signals, which makes them particularly suitable for FDM
data processing. Very short constant-time periods can be used without sacrificing the
resolution, leading to better sensitivities, especially for fast relaxing proteins.
It was also realized that various spectral representations could be directly com-
puted without diagonalizing the data matrices [77, 78]. The new expression was
named Regularized Resolvent Transform (RRT). RRT corresponds a direct, nonlin-
ear transformation of time domain signal into frequency domain spectrum. It is a
very efficient method with appealing transparency and simplicity. RRT is also quite
flexible and can be used to construct many non-FT types of spectral estimations.
In the rest of the chapter, the theory of 1D FDM will be first described in Sec-
tion 2.2, then extended to a multi-dimensional case in Section 2.3. Various ways to
regularize FDM will be discussed in Section 2.4, followed by detailed descriptions of
RRT in Section 2.5. Several ways for checking the validity of FDM/RRT results will
be discussed in Section 2.6. Finally, some conclusions are given in Section 2.7.
45
2.2 One-Dimensional FDM
In this section, we first review the theory of 1D FDM with emphasis on its two most
important properties: solving the nonlinear fitting problem by pure linear algebra and
local spectral analysis by using a Fourier-type basis. FDM is able to fit both large and
complicated signals to the summation of Lorentzian lines in an efficient and stable
way. In the case of of noisy data, the original FDM algorithm might yield a line list
that contains unphysical poles with both large amplitude and negative linewidth. By
using a more sophisticated Multi-Scale Fourier basis [79], this kind of poles can be
eliminated and a line list more consistent with the decaying signal can be obtained.
2.2.1 Nonlinear Fitting by Linear Algebra
The basic object of 1D FDM is to fit a given complex time signal c(n) = c(tn),
defined on an equidistant time grid, tn = nτ, n = 0, 1, 2, . . . , N − 1, to the sum of
exponentially damped sinusoids (i.e., Lorentzian lines),
c(n) =K∑
k=1
|dk|eiθe−in2πτfke−nτγk =K∑
k=1
dke−inτωk , (2.1)
with a total of 2K unknowns: the K complex amplitudes dk = |dk|eiθ and K complex
frequencies ωk = 2πfk − iγk. The real part of complex frequency, 2πfk, gives the line
position and negative of the imaginary part, γk, gives the line width. The absolute
value of complex amplitude, |dk|, corresponds to the integral of the peak and the
phase angle, θ, defines the phase of the peak. The line list ωk, dk corresponds to
a parametric representation of the time signal, c(n), of which each pair of (ωk, dk)
defines a peak, or, a pole.
46
Although the fitting problem of Eq. 2.1 is highly nonlinear, its solutions can be
obtained by pure linear algebra. In FDM, we assume that the complex time signal,
c(n), is associated with a time autocorrelation function of a dissipative dynamical
quantum system that is described by an effective non-Hermitian Hamiltonian operator
Ω and some initial state Φ0 [24],
c(n) =(
Φ0
∣
∣
∣Φn
)
=(
Φ0
∣
∣
∣e−inτΩ∣
∣
∣Φ0
)
. (2.2)
Here a complex symmetric inner product2 is used, (a|b) = (b|a) without complex
conjugation. We use the round brackets to distinguish it from the Hermitian inner
product, < a|b >=< b|a >∗. Ω is a complex symmetric operator,
(Ψ|Ω|Φ) = (Ψ|Ω|Φ) = (Ψ|Ω|Φ) . (2.3)
Assuming that Ω is diagonalizable, it can be written in its spectral representation,
Ω =∑
k
ωk|Υk)(Υk| , (2.4)
where ωk are the eigenvalues and Υk, the corresponding eigenvectors. Υk are or-
thonormal with respect to the complex symmetric inner product, i.e.,
(Υk|Υk′) = δkk′ . (2.5)
Inserting Eq. 2.4 into Eq. 2.2, we can recover the HIP equation of Eq. 2.1 with,
d1/2k = (Υk|Φ0) . (2.6)
Therefore the harmonic inversion problem of Eq. 2.1 becomes equivalent to the linear
algebraic problem of diagonalizing the “Hamiltonian” Ω. Or equivalently, we can
2See Appendix at the end of this chapter for a brief overview of non-Hermitian quantum mechanicsfor describing dissipative systems.
47
diagonalize the evolution operator U ≡ e−iτΩ:
U |Υk) = uk|Υk) , (2.7)
with eigenvalues uk = e−iτωk . From the eigenvalues of U , we can determine ωk, line
positions and widths, and from the eigenvectors, dk, line integrals and phases.
Even though neither U nor Φ0 is explicitly available, their matrix representation
in an appropriately chosen basis set can be computed purely from the time signal
c(n). The basis can be chosen in many different ways. The total number of basis
functions is determined by the size of the available signal. For a signal of N complex
points, the maximum basis size is N/2: each Lorentzian line requires two complex
parameters to specify it, so that at most M = N/2 lines can be uniquely fitted to a
signal of length N .
Krylov Basis
The simplest basis would correspond to a set of Krylov vectors |Φn), n = 0, 1, . . . , M−
1, with M = N/2, generated by propagating the initial state, |Φ0), using the effective
evolution operator U ,
|Φn) = Un|Φ0) = exp(−inτ Ω)|Φ0). (2.8)
The matrix elements of U in this basis are trivial to obtain as
U (1)nm =
(
Φn|U |Φm
)
=(
Φ0|UnUUm+1|Φ0
)
= (Φ0|Φn+m+1) = c(n + m + 1) . (2.9)
Since the Krylov basis is not orthonormal, the overlap matrix
U (0)nm = (Φn|Φm) = (Φ0|Φn+m) = c(n + m), (2.10)
48
must also be computed. Here convenient notations U (1)and U (0) are adopted to rep-
resent the evolution matrix and the overlap matrix. Note that both matrices are
symmetric, which is a direct result of the complex symmetric inner product formal-
ism. The fitting problem of Eq. 2.1 is then cast into a complex Generalized Eigenvalue
Problem (GEP),
U(1)Bk = ukU(0)Bk . (2.11)
Note that the eigenvectors Bk are normalized with respect U(0) as,
BTk U(0)Bk′ = δkk′. (2.12)
The complex frequencies and complex amplitudes are then,
ωk =i
τln(uk)
d1/2k =
M−1∑
n=0
[Bk]n c(n) , (2.13)
where Eq. 2.13 follows from Eq. 2.6 by substituting
|Υk) =M−1∑
n=0
[Bk]n |Φn) . (2.14)
Note that the maximum basis size M = N/2 is always used. There are several
advantages to this. First, we can thus avoid the problem of estimating the number
of resonances present in the signal, which is problematic for most realistic signals.
Second, all the available data are used to construct the evolution matrix and overlap
matrix. Therefore, the resulting data matrices contain all the information that we
can use. The rest of the problem becomes a pure mathematical problem of solving
the generalized eigenvalue problem. When the true number of resonances is less than
49
M , the matrices become singular, or near singular in the case of noisy data. In these
cases, eigenvalue solvers that are capable of handling ill-conditioned matrices should
be used, such as QZ [80] and GUPTRI [81, 82]. Numerical experiments showed that
the U matrices in 1D were actually well-behaviour enough so that stable solutions
of GEP could be readily obtained by most standard eigenvalue solvers such as those
from the LAPACK [83] library.
Armed with Eq. 2.9, Eq. 2.10 and Eq. 2.11, we have a simple method for solving
the nonlinear fitting problem of Eq. 2.1 by pure linear algebra of solving a general-
ized eigenvalue problem. The latter is a very well studied problem and an unique
solution is always guaranteed. Unfortunately, while the problem is certainly easy to
set up, it could be very difficult to solve in practice. The size of the data matrices
is determined by the signal size. For large signals, solving the generalized eigenvalue
problem becomes very demanding on the computational power as well as the com-
puter memory for storage. Large matrices might also lead to numerical problems
such as round-off error and overflow. In addition, if the signal happens to contain far
less than M = N/2 peaks (K M), then the basis becomes over-complete and the
matrices are ill-conditioned. As a result, formulated in this way, we end up solving a
large and ill-conditioned linear system, of which stable solutions are very difficult, if
not impossible, to obtain numerically. This is also one of the main drawbacks of all
full space method such as LP and the matrix pencil method we discussed in Chapter
1. In conclusion, the simple formalism with the Krylov basis has severe limitations
and cannot be applied on a regular basis to signals of size more than, say, a few
thousand data points.
50
2.2.2 Local Spectral Analysis: Fourier-Type Basis
The primitive Krylov basis leads to large and ill-conditioned linear systems for
realistic experimental signals, and therefore is not a practical choice. There is a way
to avoid diagonalizing the huge matrices in single step. By applying a special unitary
transformation to the data matrices U(1) and U(0), the diagonalization can be carried
out in multiple steps. In other words, instead of trying to analyze the whole spectrum
at one time, we can divide it into small spectral windows and analyze them separately.
This local spectral analysis is the second important property of FDM.
Any linear combinations of the primitive Krylov basis functions can also serve as
a basis. Among them, a good choice is a Fourier transform of the Krylov basis, of
which a particularly simple and efficient variant is the rectangular window Fourier
basis [53], defined as 3:
|Ψj) =M−1∑
n=0
einτϕj |Φn) , (2.15)
with ϕj being a set of equidistant values taken inside a small frequency window
of interest. For this choice the transformation from the Krylov basis Φn , n =
0, 1, 2, ..., M − 1, to the Fourier basis Ψj , j = 1, 2, ..., M , is unitary. More im-
portantly, due to the Fourier transform, each basis function |Ψj) is localized in the
frequency domain, i.e., it is a linear combination of only those eigenfunctions |Υk)
of Ω for which ωk ∼ ϕj. This implies that we can consider a small subset of, say,
Kwin M values ϕj in the frequency region [ωmin, ωmax] where the eigenfunctions
3Other window functions can also be used in constructing the Fourier basis. However, they aremathematically equivalent to the case of no window function, and practically lead to little numericaldifference. See Appendix III for more details.
51
~
window
ϕjFrequency Grid
Figure 2.1: An example of FDM single window calculations. Using a small set ofFourier-Type basis functions inside the specified frequency range, the resulting datamatrices contain mostly the spectral information of the local spectral window. Thesignal is a 1D NMR signal of human progesterone with N=2000 points and spectralwidth of 4 ppm.
can be sufficiently represented by,
|Υk) =∑
ωk∼ϕj
[
Bk
]
j|Φj) . (2.16)
Note that a tilde (∼) is added to notation B because |Υk) is now expanded in a
Fourier-type basis. It will also be added to notation U, the matrix representation of
U operator in the Fourier basis.
Eq. 2.16 means that the operator U can be diagonalized in the Fourier subspace
that corresponds to some pre-specified frequency window to yield the eigenvalues and
eigenvectors, and subsequently, the spectral parameters for that window. Figure 2.1
illustrates such a window calculation. Note that the extraction of the complex fre-
52
quencies ωk to a high precision only requires that the local completeness condition,
ρ(ϕj) =Mτ
2π≥ ρ(ωk) , (2.17)
be satisfied for the densities of the grid points ρ(ϕj) and that of the eigenfrequencies
ρ(ωk). In other words, the number Kwin of the basis vectors Φj should be larger than
the number of the eigenvalues in the small interval [ωmin, ωmax]. Moreover, the local
spectral analysis to certain extent is insensitive to the spectral properties outside
the chosen small spectral domain. Therefore, FDM has a local convergence property.
Even when the signal is so short that some crowded regions can not be fully resolved,
accurate results can still be obtained for spectral regions that are less crowded. This
property will be demonstrated with more examples when we discuss 2D FDM ant its
applications.
The idea of local spectral analysis was not first invented in FDM. The idea of
making high-resolution spectral analysis local, called “beamspacing” [84] in digital
signal processing literature, has been known for a long time and widely exploited
in various algorithms such as MUSIC and ESPRIT [3]. In NMR data processing
literature, there is also a variant of the beamspace idea that is called LP-ZOOM [50].
LP-Zoom is a modified linear prediction algorithm which uses z-transform to zoom
in on a small spectral region in the frequency domain and thus reduces the size of
the problem. In principle, it shares the many advantages of local spectral analysis
with 1D FDM. However, other difficulties of LP, such as estimating the prediction
order and accurate computation of the amplitudes (see Section 1.3.4 for more details),
remains unaddressed.
53
The matrix elements of the evolution operator in the Fourier basis can also be
evaluated purely in terms of the time signal c(n):
[
U(p)]
jj′=
(
Ψj
∣
∣
∣U (p)∣
∣
∣Ψj′
)
=M−1∑
n=0
M−1∑
n′=0
einτϕjein′τϕj′
(
Φn
∣
∣
∣U (p)∣
∣
∣Φn′
)
=M−1∑
n=0
M−1∑
n′=0
einτϕjein′τϕj′ c(n + n′ + p) (2.18)
=M−1∑
n=0
M−1∑
n′=0
ei(n+n′)τϕjein′τ(ϕj′−ϕj) c(n + n′ + p) . (2.19)
The double sum in Eq. 2.18 can be simplified to a single sum by first changing the
variables from (n, n′) to (l = n + n′, n′), Eq. 2.19, and breaking it into two terms,
M−1∑
n=0
M−1∑
n′=0
=M−1∑
l=0
l∑
n′=0
+2(M−1)∑
l=M
M−1∑
n′=l−(M−1)
,
then analytically evaluating the summations over n′, which gives,
[
U(p)]
jj′=
1
1 − eiτ(ϕj′−ϕj)
M−1∑
n=0
einτϕjc(n + p)
− eiτ(ϕj′−ϕj)M−1∑
n=0
einτϕj′ c(n + p)
+ e−i(M−1)τ(ϕj′−ϕj)2(M−1)∑
n=M
einτϕj′ c(n + p)
− eiMτ(ϕj′−ϕj)2(M−1)∑
n=M
einτϕjc(n + p)
. (2.20)
Note that the final result Eq. 2.20 is symmetric with respect to the interchange of
indices j and j′ as it should be. We can rewrite Eq. 2.20 in a compact form that can
be easily generalized to the multi-dimensional cases,
[
U(p)]
jj′= S
∑
σ=0,1
eiσ[τM(ϕj′−ϕj)+π]
1 − eiτ(ϕj′−ϕj)(2.21)
×(σ+1)(M−1)
∑
n=σM
einτϕjc(n + p) ,
54
where S defines symmetrization operator over the indices j and j′:
S gjj′ = gjj′ + gj′j . (2.22)
In principle, Eq. 2.21 is correct for all choices of ϕ and ϕj′ except for the singularity
arising at ϕj = ϕj′. To obtain a numerically practical expression for this singular
case, we evaluate the ϕj → ϕj′ limit, which leads to,
[
U(p)]
jj=
2(M−1)∑
n=0
einτϕj (M − |M − n − 1|)c(n + p) . (2.23)
Notably and quite importantly, the resulting matrices U(p)
have a sinc-like struc-
ture with generally large diagonal and decaying off-diagonal terms. The latter become
much smaller than the former once Mτ |ϕj −ϕj′| 2π. It is this structure that justi-
fies the possibility to perform the eigenvalue calculation in a small Kwin ×Kwin block
fashion for possibly large M × M matrices U(p)
.
After the new matrix representations are computed, the same generalized complex
eigenvalue equation of Eq. 2.11 can be solved. From the eigenvalues and eigenvectors,
we can compute the complex frequencies and complex amplitudes. According to
Eqs. 2.6 and 2.15 the amplitudes dk are given as4,
d1/2k = B
T
k C , (2.24)
where BT
k is the transpose of eigenvectors, and the 1 × Kwin column vector C is the
FT of the original 1 × M signal array C:
[
C]
j=
M−1∑
n=0
einτϕjc(n) , j = 1, 2, ..., Kwin . (2.25)
4There exists an alternative expression for computing the amplitudes, which is more accurate fornarrow poles in the case of high signal to noise ratio. See Appendix II for details.
55
2.2.3 FDM as a High-Resolution Spectral Estimator
Up to this point, FDM seems to be just a parameter estimator which can provide
the parameter representation wk, dk for a given FID by fitting it into Eq. 2.1.
However, FDM can also be used as a spectral estimator. Once the parameters are
available, one can easily construct the spectral representation I(w), formally defined
as the infinite time discrete Fourier Transform (DFT) summation,
I(ω) = τ∞∑
n=0
(
1 − δn0
2
)
C(n)eiωnτ , (2.26)
where the term(
1 − δn0
2
)
multiplies c0 by 12
to correct the error introduced by the
discrete sum approximation of the continuous half-line Fourier integral (so called
“first point correction”). By substituting the damped sinusoids model Eq. 2.1 into
Eq. 2.26 and evaluating the geometric summation analytically, we can obtain the
corresponding spectral estimate, which we called the FDM ersatz spectrum,
Iτ (ω) = τK∑
k=1
dk
(
1
1 − eiτ(ω−ωk)− 1
2
)
. (2.27)
In FDM, some poles with extremely small imaginary part may occur (either due to
the present of noise or real peaks), which leads to high spikes in the ersatz spectrum
producing unfavorable results. Therefore, it is often useful to include a smoothing
parameter Γ > 0 to improve the appearance of the spectra,
Iτ (ω) = τK∑
k=1
dk
(
1
1 − eiτ(ω−ωk+iΓ)− 1
2
)
. (2.28)
The absorption mode ersatz spectrum then corresponds to,
Aτ (ω) = Re[Iτ (ω)] = τK∑
k=1
Re[
dk
(
1
1 − eiτ(ω−ωk+i Γ)− 1
2
)]
. (2.29)
56
In a single calculation, FDM can only obtain results for a small frequency window.
The construction of the spectral estimation for a larger spectral range requires com-
bining the results from multiple window calculations. To reduce the inaccuracies at
the window edges, the adjacent windows overlap with each other, typically by 50%.
We first construct segments of the spectrum for each windows, then sum them up
with appropriate weighting functions, such as cosine square functions, that add up to
one,
Iτ (ω) =∑
c
gc(ω)Iτc (ω) , (2.30)
where the subscript c represents different spectral windows, and the weighting func-
tions gc(ω) satisfy∑
c gc(ω) = 1. Figure 2.2 illustrates such a multi-window imple-
mentation of FDM. Cosine square functions (shown as dotted lines) are used to weight
the segments before they are concatenated together, so that the center parts of each
window segments will contribute more to final spectrum than the edges. Note that
for windows on the two edges the large spectral range of interest, only half of the
corresponding segments of the spectrum are retained.
Practical Issues on Spectral Estimation via FDM
There are several additional complications involved in constructing a stable spec-
tral representations using the spectral parameters computed by FDM.
1. Iτ (ω) or I(ω)?
For a very long time after the invention of FDM, a spectral representation I(w)
that corresponds to the infinite time Fourier integral had been used. Substituting
Eq. 2.1 into the infinite time FT integral (see Eq. 1.1) and evaluating the integration
57
Fourier-Type Basis
win. 1
win. 5
win. 4
win. 3
win. 2
Figure 2.2: A multi-window implementation of FDM. Cosine-square functions, shownas dotted lines, are used for weighting the segments of the spectrum before they areconcatenated together.
58
analytically yields,
I(ω) = iK∑
k=1
dk
ω − ωk. (2.31)
Generally, for small τ we have I(ω) ≈ Iτ (ω), but unlike Iτ (ω), which is periodic in ω
with the period equal to the Nyquist width 2π/τ , I(ω) is not periodic. However, as we
are dealing with discrete time signals, the resulting spectra should be periodic. When
there are some broad poles such that γkτ << 1 is not satisfied, I(ω) becomes a bad
approximation to the DFT spectrum. The mutual interferences and cancellations of
the poles are not conserved, leading to some instabilities in the baseline. Numerical
experiments showed that broad poles occurred frequently when FDM was applied
to analyze noisy signals and/or signals with a strong background. For example,
Figure. 2.3 compares the FDM ersatz spectra obtained using Eq. 2.31, trace (b), and
Eq. 2.27, trance (c). While the Iτ (ω) spectrum is very stable, the I(ω) spectrum
shows significant baseline distortions. This demonstrates that Eq. 2.31 is dangerous
to use if there are interfering poles ωk ≡ νk − iγk with both large widths γk and large
amplitudes dk. With the slightly wrong formula (Eq. 2.31 rather than Eq. 2.27) the
negative contributions are not correctly canceled by the positive ones, leading to the
baseline distortions. Similar discussions were also independently given before in the
nonlinear least squares fitting literature [30]. In conclusion, Iτ (ω) should always be
constructed.
2. What should we do with unphysical poles with negative γk?
Due to the imperfections of the signals such as noise and lineshape distortions,
and certain limitations of FDM, poles with negative γk (or negative linewidth) often
59
appear in the line list. This formally corresponds to unphysical exponential increasing
signals. Now we are faced with a dilemma on what to do with such kind of poles.
There are a few options.
a. Do nothing.
b. Simply flip the sign of all negative γk.
c. Throw the unphysical poles away.
This dilemma also occurs in linear prediction. In LP, the amplitudes dk are com-
puted by solving a least squares problem after the frequencies ωk are calculated by
rooting a polynomial. An ωk with negative γk can blow up all the amplitudes. Thus,
generally, a combination of (b) and (c) has to used: the ωk with small γk are retained
with the sign of γk flipped, while the ωk with large and negative γk values are re-
jected. Even when LP is only used to extrapolate the signal, it is always necessary
to first root the characteristic polynomial and then eliminate roots corresponding to
exponentially growing signals, in order to obtain a stable, meaningful extrapolation.
In FDM the situation is quite different as the amplitudes dk are computed simul-
taneously with the frequencies ωk using the eigenvectors and eigenvalues from the
same generalized eigenvalue problem. Therefore, γk < 0 does not necessarily lead to
any numerical instability in FDM. In the early applications of FDM [53, 67], I(ω)
was always constructed. A narrow peak with negative γk will appear upside-down in
the absorption spectrum, which can be easily fixed by flipping the sign of γk. How-
ever, because of the local nature of the Fourier basis, which is manifestly incomplete,
some computed pairs (ωk, dk) may have large and negative γk and large dk values.
When I(ω) is constructed, these entries would often result in a noticeable baseline
60
distortion, as demonstrated by trace (b) of Figure. 2.3. Neither throwing away such
a “spurious” entry nor flipping the γk would fix the baseline.
2.12.22.32.42.52.62.7
a). I(τ)
(ω) DFT
b). I(ω) FDM
c). I(τ)
(ω) FDM
d). I(ω) Multi-Scale FDM
1H Chemical Shift / ppm
Figure 2.3: 1D proton NMR absorption spectra of progesterone for a representativespectral region of a noisy signal. The signal length is N = 2000 and spectral widthis SW = 4 ppm (2 KHz). All the entries returned by FDM were used to reconstructthe ersatz spectrum without any attempt to identify and/or throw away any spuriouspoles. From the top to the bottom, the traces are,(a). DFT with appropriate apodization function.(b). FDM ersatz spectrum computed using Eq. 2.31 with single-scale FDM.(c). FDM ersatz spectrum computed with Eq. 2.27 with single scale FDM.(d). FDM ersatz spectrum computed using Eq. 2.31 Eq. 2.31 with Multi-Scale FDM.
Trace (c) of Figure 2.3 was a little unexpected in that Iτ (ω) constructed with
the line list ωk, dk containing some “spurious” entries leads to the correct spectrum
with undistorted baseline. It can be explained by the fact that it is the Iτ (ω), not
61
I(ω), which is invariant to the basis set representation. Due to the existence of
very delicate mutual interferences and cancellations of the contributions all poles in
Eq. 2.27, the instabilities do not occur if one keeps all the entries produced by FDM.
This also explains the stability of the RRT (see Section 2.5) which relies on the same
DFT based expression. Clearly, if Iτ (ω) gives the correct baseline in the presence
of such poles, I(ω) must fail as the two expressions are quite different for large γk.
Accordingly, when possible,
• All the poles with significant amplitudes dk must be retained in the sum.
• Only narrow poles with γk < 0 should be flipped: γk → −γk.
Note that even though the spectrum Iτ (ω) looks correct, the underlining broad
poles with negative γk and large dk do not correctly describe the time domain data.
Therefore, for obtaining a line list consistent with the decaying time signal, and with
minimized cancellation effects, the use of Multi-Scale FDM is advantageous, which
will be described in the next section.
A Numerical Example
To demonstrate the ability of FDM to identify and resolve Lorentzian lines with
a wide range range of linewidths and splittings, a model signal named “Jacob’s Lad-
der” [67] is constructed. The signal contains 50 triplets with progressively smaller
linewidths and splittings from high frequency to low frequency,
c(n) =50∑
k=0
e−i2π×(0.001n)×(500.0−1.0i) 0.9k
+
2 e−i2π×(0.001n)×(497.5−1.0i) 0.9k
+ e−i2π×(0.001n)×(495.0−1.0i) 0.9k
(2.32)
62
A total of N = 64, 000 complex data points were produced. Figure 2.4 shows the
spectrum of the model signal. Due to the wide range of linewidths and splittings,
accumulation of lines at low frequency and large size of the signal make it a very chal-
0 100 200 300 400 500
~
Frequency (Hz)5
Frequency (Hz)
10
Figure 2.4: A total of 50 triplets: beginning with 1.0 Hz linewidth and 2.5 Hz couplingcentered at 497.5 Hz and finishing at 5.73 mHz linewidth and 14.3 mHz couplingcentered at 2.82 Hz. The spectral width is 1 KHz. The time signal contains 64,000data points. The wide range of linewidth and splittings, accumulation of lines nearzero frequency, and large size make this signal challenging for non FT methods.
lenging signal for any other non-FT methods such as Linear Prediction and Maximum
Entropy Reconstruction.
Figure 2.5 compares the DFT and FDM on the densest part of the spectrum of the
test signal, using 32K and 64K data points. It demonstrates the ability of FDM to
fit both large and complicated time domain signals to the summation of Lorentzians,
63
and provide resolution beyond the FT uncertainty principle. Due to the local spectral
analysis, it took only a few seconds to obtain these results on a small PC workstation.
2.5 3 3.5 4 4.5 5
Nsig=64K
FDM
DFT
Nsig=32K
FDM
DFT
Frequency (Hz)
ExactSpectrum
Figure 2.5: A comparison of FT and FDM on the densest part of the signal “Jacob’sLadder” (Figure 2.4). In FT calculations, the signal was first zero-filled to 128K datapoints. FDM resolves all the multiplets with signal length of 32K points, while theleft most triplets are not resolved by FT even with 64K data points.
A Case Where FT Beats FDM
We have demonstrated that FDM is able to effectively use the information that
the signal contains purely Lorentzian lines to provide resolution beyond the FT uncer-
tainty principle. However, this does not mean that FDM will provide better resolution
for all Lorentzian signals. It is most advantageous for FDM when the distribution of
64
the peaks is highly non-uniform, and highly disadvantageous for FDM when the peaks
are uniformly distributed. To demonstrate the point, a special model signal that con-
tains 21 peaks evenly distributed in the center part of the spectrum is constructed.
-2 -1 0 1 2
DFT
-2 -1 0 1 2
FDM
Frequency (Hz) Frequency (Hz)
N = 40
N = 60
N = 80
N = 100
N = 40
N = 60
N = 80
N = 100
Figure 2.6: A special case where DFT can beat FDM even though the signal containspurely Lorentzian lines. The model signal is integerized with the first point c0 = 2048.
The signal is integerized to mimic ADC with the first point being c0 = 2048. Even
though this signal contains purely Lorentzian lines, it is optimal for DFT processing,
because DFT provides uniform resolution and thus works the best when the features
are uniformly distributed. Figure 2.6 compares the performance of DFT and FDM on
this signal. It shows that DFT is able to resolve all 21 peaks using as few as N = 60
data points, while FDM does not converge until N ≥ 100. However, note that even
65
though DFT is able to resolve all 21 peaks for N ≥ 60, the linewidths converge very
slowly, while FDM is able recover all parameters very accurately once the size of the
signal is above certain threshold.
Summary of the FDM Algorithm
The practical numerical procedure for implementing the FDM algorithm can be
summarized as the following steps:
1. Choose a spectral window: specify a small enough frequency range of interest,
[ωmin, ωmax], inside the Nyquist range for a given time signal of length N , cn =
c(nτ), n = 0, 1, . . . , N − 1. “Small enough” is operationally defined by the resulting
size of the linear algebraic problem, which could depend on the type of the signal and
computational power available.
2. Set up the Fourier basis: set up an evenly spaced angular frequency grid inside
the specified window, ωmin < ϕj < ωmax, j = 1, 2, . . . , Kwin, where the basis size Kwin
is determined by Kwin = ρ M(ωmax − ωmin) τ/2π. M = N/2 is the maximum size of
basis. ρ ≥ 1 is the basis density, defined in Eq. 2.33. Typically ρ = 1.1 ∼ 1.2.
3. Compute the matrix elements using Eqs. 2.21 and 2.23. The resulting matrices,
U(1)
and U(0)
, are complex symmetric and of size Kwin × Kwin.
4. Solve the generalized eigenvalue problem of Eq. 2.11 using any standard eigen-
value solvers such as CG [83] and QZ [80] routines.
5. Compute the complex amplitudes according to Eq. 2.24.
6. Use the line list ωk, dk as an input for the spectral estimation (optional).
7. If of interest, go to step (1) choosing the next frequency window.
66
Note that in the multi-window implementation, large spectral range can be spec-
ified, which is then automatically divided into multiple smaller windows.
Computational Cost of 1D FDM
The most time consuming steps are the matrix construction, step (3), and diago-
nalization, steps (4). Close inspection of Equations 2.21 and 2.23 shows that for an
arbitrary choice of 1D grid of size Kwin, the evaluation of all U(p)
elements scales as
Kwin × N : linear with respect to the size of the time signal and the basis size. The
cost of the diagonalization step scales as K3win. Therefore the overall computational
cost of FDM is a Kwin × N + bK3win, where a and b are scaling factors. In the case
of small spectral windows, K2win << N , the speed determining step is the U matrices
evaluation step, which is often the case for 1D FDM. In order to process the whole
spectral range, approximately N/Kwin window calculations are needed. Thus, the
total cost of processing a 1D signal roughly scales as (N/Kwin) × (KwinN) = N2, if
small windows are used. Therefore, the computational cost of FDM is comparable
to that of Discrete FT, although still more expensive than N log(N) of FFT. For
example, for a signal of 10000 complex data points, FDM takes a few seconds to
analyze the whole spectral range using a basis size of 50 on single AMD XP 1800+
workstation, which cost less than $500 in May of 2002.
2.2.4 Multi-Scale Fourier-Type Basis
As mentioned above, unphysical poles with large and negative linewidths and
large amplitudes can occur in FDM, manifested as the baseline instabilities of the
67
I(ω) spectrum. This can be due to the narrow band Fourier basis functions, Eq. 2.15,
which are highly localized in the frequency domain. On one hand, localization is a
big advantage of using the Fourier basis, on the other hand, in the presence of noise,
a narrow band basis is inadequate to represent a broad spectral feature that is not
localized in a single window, as each basis function |Ψj) is “locked” on the noise peaks
that appear close to ϕj and cannot “see” the peaks that are far away. For example, if
there exists a broad peak that centers outside of the current spectral window but tails
into the window, the narrow band Fourier basis would not be enough to accurately
represent the contribution of this broad peak. Thus, with the narrow band Fourier
basis only, there might be some ambiguity in reproducing the broad spectral features
or the baseline, and this is the source of the instability. Obviously, it can sometimes
be reduced by using larger windows. However, increasing the basis size does not
always work, and even when it does, it is impractical due to the unfavorable scaling
of the computational cost for an eigenvalue solver. In this section, we will introduce
a more sophisticated way of local spectral analysis using a Multi-Scale Fourier-type
basis. The improved algorithm, named Multi-Scale FDM, is more robust and reliable,
and at the same time computationally efficient.
The idea is illustrated in Figure 2.7. In addition to the narrow band Fourier basis
functions inside the window, we add some coarse (i.e., delocalized or broad band) basis
functions outside of the window, so that the global behaviour of the spectrum can
be captured, and its interference on the window calculations minimized. Since these
global features such as broad peaks and background do not require high resolution
we do not need many of such coarse basis functions. We also want to minimize the
68
2.40 2.30 2.20 2.10 2.00 ppm
ϕ j
low resolution
~
high resolution low resolution
<--
<-- <--
window
~
Figure 2.7: An example of a multi-scale basis set and the spectrum obtained usingthis basis for the same signal used in Fig. 2.1. Kwin = 10 narrow band and Kc = 20coarse basis functions (indicated by an impulse at each ϕj) were used. The coarsefunctions are distributed non-uniformly according to the displacement from the edgesof the window (see text).
number of coarse basis functions with the condition that the non-localized features
that may affect the local spectral analysis are represented adequately, which can be
achieved by choosing the most efficient coarse basis distributions.
Given that the farther away from the current window, the less significant the effect
of a broad spectral feature to the local analysis, the coarse basis could be chosen such
that the spacing between the adjacent coarse basis functions and, accordingly, their
bandwidth, monotonically increase with respect to the distance between the center
of the basis function (ϕj) and the spectral window of interest. Thus only the features
which are broad enough to affect the current window will be captured by the coarse
69
basis. Here the local density of the basis functions is defined as
ρj =2∆ϕmin
|ϕj+1 − ϕj−1|(2.33)
where ∆ϕmin = 2π/τMmax is determined by the spacing between the narrow band
Fourier basis functions. The Fourier length should be scaled accordingly to the local
density using Mj = ρjMmax, so that the bandwidth of the basis functions is consistent
with the basis density. For a given small spectral window we can construct a multi-
scale basis, which contains Kwin narrow band Fourier basis functions localized in the
window and Kc coarse basis functions spread over a wide spectral range:
|Ψj) =Mj−1∑
n=0
einτϕj |Φn), j = 1, 2, . . . , K = Kwin + Kc , (2.34)
with the Fourier length Mj depending on j: Mj = Mmax for the narrow band, and
Mj = Mmax ρj , for the coarse basis functions.
The elements of U matrices can be then evaluated in terms of the time signal.
Using the definition of multi-scale FT basis functions |Ψj) and |Ψj′), Eq. 2.34, and
the assumption of Eq. 2.2, we obtain,
[
U(p)]
jj′=
Mj−1∑
n=0
Mj′−1∑
n′=0
einτϕjein′τϕj′ c(n + n′ + p)
=Mj−1∑
n=0
Mj′−1∑
n′=0
ei(n+n′)τϕjein′τ(ϕj′−ϕj)c(n + n′ + p) . (2.35)
Unlike previous case of single-scale Fourier basis, here Mj and Mj′ do not always
equal to each other, making the expression a little more complicated. However, using
a similar strategy used to derive the efficient expression of Eq. 2.21, we can still
reduce the numerically expensive double sum to several single summations. First, by
70
substituting l = n + n′, and assuming Mj < Mj′, we break it into three terms,
Mj−1∑
n=0
Mj′−1∑
n′=0
=Mj−1∑
l=0
l∑
n′=0
+
Mj′−1∑
l=Mj
l∑
n′=l−(Mj−1)
+
Mj′+Mj−2∑
l=Mj′
Mj′−1∑
n′=l−(Mj−1)
.
Then by evaluating analytically the summations over n′, we can replace each term by
single summations. After some manipulations, an efficient expression for off-diagonal
matrix elements results, which is given in a compact form here,
[
U(p)]
jj′= S
∑
σ=0,1
eiσ[Mj′τ(ϕj′−ϕj)+π]
1 − eiτ(ϕj′−ϕj)×
σ(Mj′−1)+Mj−1∑
n=σMj′
einτϕjc(n + p) , (2.36)
where S is the symmetrization operator over the subscripts j and j′, as defined in
Eq. 2.22. For the diagonal matrix elements, ϕj = ϕj′, we have,
[
U(p)]
jj=
2Mj−2∑
n=0
(Mj − |Mj − n − 1|)einτϕjc(n + p) . (2.37)
Note that when Mj = Mj′, Eqs 2.36 and 2.37 boil down to the formulas previously
derived for the case of single-scale Fourier basis, Eqs 2.21 and 2.23.
Once the U-matrices are available in this new basis, the generalized eigenvalue
problem Eq. 2.11 can be solved. Due to Eqs. 2.13 and 2.34 the complex amplitudes
are computed as,
√
dk =K∑
j=1
[Bk]j
Mj−1∑
n=0
einτϕjc(n) . (2.38)
Figure. 2.7 shows an example of such a multi-scale basis set for a particular win-
dow together with the FDM ersatz spectrum obtained using this basis and plotted,
intentionally, in a wide spectral range. As expected, fine features are captured inside
the window where dense and narrow band basis functions are used, and only coarsely
resolved features appear outside this window. Furthermore, the spectral resolution
71
decreases smoothly in the directions away from the window, so there are no edge
effects associated with the local spectral analysis. This makes it easier to combine
the results of different windows to construct the overall spectrum. More importantly,
with the presence of coarse basis, the line list is free from unphysical poles with large
and negative γk and large amplitudes, although poles with small and negative γk
might still occur. The latter can be easily fixed simply by flipping the sign. With
Multi-Scale basis, the spectral representation I(ω), defined in Eq. 2.31, will also be
stable and baseline distortion free. An example is shown in Figure 2.3, trace (d).
However, Iτ (ω) is still preferred and should be used.
Some Aspects of Numerical Implementations
A comment must be made on how to compute the discrete Fourier sums for all j
and j′ in Eq. 2.36 efficiently. The first two sums need to be computed only once for
every j. The other two sums depend on both indices j and j′, and might seem expen-
sive. Here we describe an algorithm which scales as Mmax × Kwin for all the matrix
elements of U(p) rather than MmaxK2win as one might think. Thus the computational
cost of evaluating the U-matrices in a multi-scale Fourier basis is similar to that of
the original version of FDM.
For convenience, we introduce the notation:
g(p)j (M) ≡
M+Mj−1∑
n=M+1
einτϕjc(n + p) .
Since g(p)j (Mmax − 1) only depends on ϕj , it can be evaluated once and stored in an
array for later use. Now given g(p)j (Mmax), one can obtain g
(p)j (Mj′) recursively for all
72
Mj′ < Mmax according to
g(p)j (M − 1) = g
(p)j (M) + eiMτϕjc(M + p)
− ei(M+Mj−1)τϕj c(M + Mj − 1 + p) .
Finally, note that there is no need to re-evaluate the Fourier sums in Eq. 2.36
for p = 1, once they have been computed for p = 0. There exists a simple recursive
relation between these two,
M∑
n=0
einτϕc(n + 1) = e−iτϕ
[
M∑
n=0
einτϕc(n) − c(0)
]
eiMτϕc(M + 1) .
A Double-Scale Fourier Basis
We would also like to find out the most efficient coarse basis distribution that re-
quires minimum computational efforts to obtain similarly reliable results. We found,
again by numerical experiment, that for the simplest realization of a multi-scale
Fourier basis, one can consider just two scales with Mj = Mmax, for the narrow
band window basis, and Mj = Mc Mmax, for the coarse basis. This corresponds to
having two equidistant grids with spacings, ∆ϕmin = 2π/τMmax and ∆ϕc = 2π/τMc,
respectively. At first look, the double scale distribution may not sound as effective as
the real “multi-scale” basis with smoothly decaying basis density. However, numer-
ical experiments showed that such a basis distribution is sufficient to obtain “good”
results for most circumstances, while there may always exist some cases that require
more complicated basis distributions.
Unlike the example shown in Figure. 2.7 which might seem a little complicated,
the double-scale basis implementation simplifies the calculation of the U-matrix ele-
73
ments. Furthermore, the size of coarse basis can be further reduced by putting coarse
basis functions |Ψj) only in the regions with significant peaks in a low resolution pic-
ture. Numerical implementation of this “adaptive coarse basis” requires pre-applying
DFT or FDM to a short signal to obtain a low resolution picture in order to decide
where the coarse basis functions should be placed. The examples shown the next sec-
tion were all obtained using the double-scale FDM (but without the adaptive basis
implementation).
Multi-Scale Fourier Basis Using Complex Frequency Grid
There is an even simpler implementation of the multi-scale FDM which does not
require any of the new formulas derived earlier in this section. Instead of using a real
frequency grid ϕj, a list of complex values ϕ(c)j = ϕj + iγj is used,
|Ψj) =Mmax−1∑
n=0
einτϕj e−nτ γj |Φn) , (2.39)
Note that if γj = 0 this new definition is the same as Eq. 2.15. When γj > 0, the
exponential e−nτ γj acts as a FT weighting function, and changes the effective FT
length. Adjusting the imaginary part of the complex grid points ϕ(c)j will have the
same effect as directly adjusting the FT length, Mj in Eq. 2.34. The relation between
γj and Mj is given as,
Mmax−1∑
n=0
e−nτ Im[ϕ(c)j
] = Mj . (2.40)
This equation can be solved analytically by replacing the summation with a integral
and solving the resulting simplified equation. The solution can then be expanded in
Taylor series and approximated by,
74
τ Im[ϕ(c)j ] =
1t+ x − x2 + 3
2x3 − 8
3x4 + 125
24x5 + O(x6) , if t < 0.685
2.82(1 − t) , if t > 0.685
, (2.41)
where t = Mj/Mmax, x = − exp(−1/t)/t. Note that Mj has the same dependence on
basis distribution as previously described. A plot of γj vs. t is shown in Figure. 2.8.
With the complex definition of the multi-scale FT basis, the formulas of the single
0 0.2 0.4 0.6 0.8 1
0
10
20
30
40
50
Mj / M
max
τ Im
[ϕj]
Figure 2.8: A plot of τ Im[ϕ(c)j ] vs. Mj/Mmax for complex frequency grid implemen-
tation of Multi-Scale Fourier basis.
scale FDM by used by simply replacing the real grid ϕj by the complex one. This
can greatly simplify the implementation of multi-scale Fourier basis in 2D or higher.
Numerical experiments showed that the performance of the multi-scale FDM with
complex grid is similar to that of the original implementation.
75
Numerical Examples
The example of Fig. 2.9 demonstrates the reliability and robustness of the multi-
4.0 3.5 3.0 ppm
FT spectrum
x8
x8
x8
Kwin=4, Kc=10
x8
Multi−Scale
FDM
FDM (1) Kwin=200, Kc=0
FDM (2)
Kwin=200, Kc=0
Figure 2.9: This example shows the spectra of an artificially made signal (see text),processed by three different methods: single-scale FDM, multi-scale FDM and DFT.The spectrum marked with FDM (2) was obtained with the same parameters asFDM (1) but the position of the windows was slightly shifted. The instability in bothrepresenting the background spectrum and resolving the fine features (the doublet)occurs even with Kwin as large as 200 basis functions (corresponding to 0.88 ppmspectral window). The doublet is not resolved in the FT spectrum either due tothe FT uncertainty principle, although the spectrum envelop is reproduced correctly.The result obtained by the multi-scale FDM with just Kwin + Kc = 4 + 10 = 14 basisfunctions is superb in all respects and requires minimal computational effort.
scale FDM. Even under extreme conditions it is still as stable as the Fourier spectrum
but can deliver higher resolution than both the Fourier spectrum and the previous
single-scale version of FDM. We considered the same NMR signal as that used in
76
Figure. 2.1 and 2.7 to which, in the region with no genuine NMR lines, we added
artificially some very broad Lorentzian lines to simulate a huge background, and
some very narrow lines. I(ω) instead of τ (ω) was used to reveal the instabilities due to
unphysical poles. For this extreme case, the single-scale FDM does not work well even
when a quite big window basis is implemented. For example, for signal length N ∼
2000 and Kwin ∼ 200 window basis functions the spectrum envelop is not reproduced
correctly. Moreover, in this case the results appeared to be quite sensitive to the
input parameters, such as a slight shift in the window position. While reproducing
the background spectrum is not a problem for the Fourier spectrum, it cannot resolve
the doublet made out of two narrow, equal in height and closely spaced Lorentzian
lines. Quite surprisingly, the multi-scale FDM with just Kwin+Kc = 4+10 = 14 basis
functions per window reproduces all the relevant spectral features quite accurately.
Moreover, the doublet is now much better resolved than in both the single-scale FDM
and FT spectra.
Finally, Fig. 2.10 presents an IR spectrum. The interferogram contained 4744 data
points that were processed by both FT and multi-scale FDM to generate the absolute
value spectra |I(ω)|. The FT spectrum is very hard to interpret and, probably, hard
to quantify by conventional means, as the peaks are not quite narrow and both the
overlapping effects and the interference with the background are significant. Unlike
the FT case, the FDM peaks are generally much sharper. Notably and most impor-
tantly, the FDM spectrum is fit by the form of Eq. 2.1, so the parameters of the peaks
(such as the positions, widths and amplitudes) are known, while fitting the absolute
spectrum |I(ω)| by Lorentzians would be a very challenging project. Also note that
77
the overall shape of the spectrum is reproduced well by the multi-scale FDM, while
the single-scale version would be very unstable for this signal because of the strong
background.
1430.0 1530.0 1630.0 cm−1
(a)
−−
>−
200.0 2200.0 4200.0
(c) FT spectrum
(b) Multi−Scale FDM
Kwin=20, Kc=30
Figure 2.10: FT-IR spectrum (a) and an interesting part of the same spectrum pro-cessed by the Multi-Scale FDM with Kwin + Kc = 20 + 30 = 50 (b) and by FT (c).The interferogram contained 4744 data points.
2.2.5 Summary
1D FDM is an efficient and stable algorithm for fitting both large and compli-
cated time domain signals to the summation of Lorentzian lines. It solves the highly
nonlinear fitting problem by recasting it into pure linear algebraic problems of diag-
onalizing some small data matrices. FDM can be used as a parametric estimator as
well as a spectral estimator. Provided that signal sufficiently satisfies the Lorentzian
78
model, FDM can deliver resolution beyond the FT time-frequency uncertainty prin-
ciple. While the spectral estimation, Iτ (ω), given by FDM is always very stable,
unphysical poles with large and negative linewidths and large amplitudes can occur
when the signal is very noisy and/or contains nonlocalized features such as strong
background and broad peaks. A multi-scale Fourier-type basis is introduced to im-
prove the efficiency of local spectral analysis and suppress these unphysical poles.
With a Multi-Scale Fourier basis, it is possible to use much smaller spectral windows
and still obtain reliable results even for very noisy signals.
Currently the 1D FDM algorithm is already a well-developed method which is very
stable, sufficiently fast, and can provide reliable high-resolution spectral estimations,
given that the signals sufficiently satisfy the Lorentzian model. However, obtaining
a compact true line list is still problematic. In the case of heavy overlapped features
and/or very noisy data, the lines obtained by FDM do not necessarily all represent
the true resonances in the time signal, even though the spectrum computed from
such a line list is a reliable estimate of the true infinite DFT spectrum. Any attempt
for non-trivial use of the line list, such as large linear phase corrections and line
narrowing, might break the mutual interference and cancellation of the lines and
result in unstable spectral estimations.
79
2.3 Multi-Dimensional Extensions of FDM
In this section, we describe multi-dimensional extensions of the FDM algorithm
(MD-FDM). As we discussed in Chapter 1, multi-dimensional spectral analysis is
much more challenging, and conventional FT spectral analysis has major limitations.
Interestingly, the true power of FDM actually lies in its multi-dimensional extensions,
in the ability to process the whole multidimensional data set together to obtain the
maximum possible resolution in all dimensions. We will start from explaining the
concept of true multi-dimensional spectral analysis in Section 2.3.1, then describe a
“naive” version of multi-dimensional FDM, which requires simultaneous diagonaliza-
tion of multiple matrices, in Section 2.3.3. In Section 2.3.4, we will describe a more
robust approach that uses the Green’s functions to compute the multi-dimensional
spectral estimations directly using the results of solving several generalized eigenvalue
problems. However, due to the ill-conditioning of the problem, spectra obtained from
single FDM calculations are often contaminated with artifacts that are sensitive to
any change of the FDM parameters and/or input signal. Several primary averaging
procedures which make use of this sensitivity to suppress these artifacts will then be
described in Section 2.3.5. Finally, a summary will be given in Section 2.3.6.
For the sake of simplicity, only 2D FDM will be explicitly discussed. Extension
of 2D FDM to higher dimensionality is straightforward and does not involve any new
fundamental difficulty.
80
2.3.1 What Is a True Multi-Dimensional Method?
Before we describe the theory of multi-dimensional FDM, let’s first define the
concept of true multi-dimensional spectral analysis. As we discussed in Chapter 1,
multi-dimensional FT is essentially a 1D spectral analysis method. The FT time-
frequency principle applies to all dimensions independently. In order to obtain high
resolution in all dimensions, FT requires long signals in all dimensions. Due to many
practical limitations such as limited experimental time and limited long-term instru-
mental stability, the multi-dimensional NMR signals are typically truncated in the
indirect dimensions, leading to poor FT resolution in the corresponding dimensions.
This is one of the major limitations of multi-dimensional FT-NMR.
On the contrary, for a true multi-dimensional method, the obtainable resolution in
all dimensions is determined together by the total information content of the signal,
which can be roughly measured by the total size of the signal, N1 × N2 × . . . × ND,
where Nl is the number of data points along the lth dimension. Figure 2.11 shows
a schematic illustration of a true two-dimensional spectral analysis method. As long
as the total size of the 2D signal is sufficiently large, whether the signal is long or
short in a particular dimension has little effect on the obtainable resolution for a
true 2D spectral analysis method. In other words, a long time dimension can be
used to enhance the resolution along an orthogonal dimension for which it may be
time-consuming, or impossible, to obtain a long signal. This property is particularly
useful for NMR data processing. In multidimensional NMR experiments, acquisition
dimension is cheap and long signals are always available, while the indirect dimensions
81
are expensive and typically truncated. Therefore a multi-dimensional NMR signal
is almost always very large and might contain enough information to pin down all
the multi-dimensional features, if we have a true-multidimensional spectral analysis
method capable of using the information effectively. Multi-dimensional FDM is such
a method, as it will become clear later in this section.
N1
N2
N2
N1
2D Signal
2DSignal
True 2D Spectral Analysis
ω2
ω1
ω2
ω1
Figure 2.11: A schematic illustration of a true two-dimensional spectral analysismethod. As long as the total area of the signal, defined as N1 × N2, is sufficientlylarge, a true 2D spectral analysis method should be able to obtain similar 2D spectralestimations no matter whether one dimension is longer or shorter than the other.
82
2.3.2 2D Harmonic Inversion Problem
Given a 2D complex valued time signal c(~n) ≡ c(n1τ1, n2τ2), with n1 = 0, 1, ..., N1−
1, n2 = 0, 1, ..., N2 − 1, where ~n is the time vector5 defined on a 2D equidistant time
grid, we want to fit it to the summation of 2D Lorentzian lines,
c~n =K∑
k=1
dke−i~n~ωk ≡
K∑
k=1
dk exp (−in1τ1ω1k − in2τ2ω2k) , (2.42)
where ~ωk ≡ (ω1k, ω2k) are vectors of unknown complex frequencies, ωlk = 2πflk − iγlk,
and dk are unknown complex amplitudes. This is a 2D Harmonic Inversion Prob-
lem (2D HIP). The total number of unknown complex parameters in the 2D line list
~ωk, dk with K entries is 3K. Note that this formalism of 2D HIP is similar to but
different from those used in 2D LP [16] and the 2D matrix pencil method [22]. In
the latter cases, models with a direct product set of frequencies, ω1k, ω2k′, dkk′, k =
1, 2, . . . , K1, k′ = 1, 2, . . . , K2, are used, so that the total number of unknowns is
K1 +K2 +K1 ×K2. However, a typical 2D NMR spectrum does not have completely
direct product pattern. Even for spectra that occur in COSY [85], NOESY [86]
and TOCSY [87] types of experiments, the direct product patterns are more or less
localized. The non-direct product model of Eq. 2.42 is actually a more general for-
malism that allows a direct-product output, while leading to a much more compact
representation for spectra without direct-product patterns.
It should also be pointed out that a true multi-dimensional method will have
the maximal advantage over 1D spectral analysis when the spectra contain minimal
direct product patterns. For spectra with a lot of direct product peak patterns such as
5Even though only the 2D case is considered, the vector notations make the extension to the caseof more than 2D straightforward.
83
COSY, NOESY, and TOCSY spectra, processing the whole data set together does not
lead to a big advantage compared to trace-by-trace 1D processing. In these cases, MD
FDM could only help to improve resolution by being an efficient and stable nonlinear
parametric method. Therefore, the most interesting applications of MD FDM in
NMR are the multi-dimensional experiments that are routinely used to obtain the
chemical shift assignment such as HSQC, HNCO(A) and others. Several examples
will be given in Chapter 3.
2.3.3 A “Naive” Version of 2D FDM
Here we describe a straightforward 2D extension of FDM that was used in Ref. [68,
69, 70]. The object of 2D FDM is to fit a given 2D discrete time signal, defined on a 2D
equidistant time grid, to the summation of Lorentzian lines, i.e., solve the 2D HIP of
Eq. 2.42. starting point. Similar to the 1D case, this highly nonlinear fitting problem
can be cast into a linear algebraic problem, or more precisely, a family of generalized
eigenvalue problems, by associating the 2D time signal to a double-time correlation
function of a fictitious dynamic system with two commuting non-Hermitian symmetric
Hamiltonians Ω1 and Ω2 [68, 70],
c(~n) =(
Φ0
∣
∣
∣e−i~n~ΩΦ0
)
≡(
Φ0
∣
∣
∣e−in1τ1Ω1−in2τ2Ω2Φ0
)
. (2.43)
The Hamiltonians are diagonalizable by the same set of eigenvectors,
Ωl |Υk) = ωlk |Υk) , l = 1, 2 , (2.44)
where the eigenvectors are orthonormal according to the symmetric complex inner
product, (Υk|Υk′) = δk,k′. Inserting their spectral representations
84
Ωl =∑
k
ωlk |Υk) (Υk| (2.45)
into Eq. 2.43, we can recover the 2D HIP equation 2.42 with
dk = (Φ0|Υk)2 . (2.46)
Therefore, the nonlinear fitting problem of Eq. 2.42 is recast into a pure linear alge-
braic problem of diagonalizing the Hamiltonian operators, or equivalently, the evolu-
tion operators, Ul ≡ e−iτl~Ωl,
Ul |Υk) = ulk |Υk) , l = 1, 2 . (2.47)
The matrix representations of the evolution operators in an appropriate basis can
be computed from the 2D time signal. The simplest choice is a set of time like
Krylov basis functions created by propagating the “initial” state using the evolution
operators,
|Φ~n) ≡ U(~n)|Φ0) = Un11 Un2
2 |Φ0) . (2.48)
The elements of the evolution matrices in Krylov basis are simply given by the 2D
signal points,
[U(~p)]~n,~n′ = (Φ~n|U (~p)Φ~n′) = c(~n + ~n′ + ~p) , (2.49)
for any time vector ~p = (p1, p2), among which for ~p = (0, 0), (1, 0), and (0, 1) U(~p)
corresponds to the overlap matrix U0 and evolution matrices in two time dimensions,
U1 and U2, respectively. The requirement of no missing entries in these matrices
limits the size of the Krylov basis to M1 × M2 = N1/2 × N2/2. Similar to 1D FDM,
the maximum basis size is always used to avoid the problem of estimating the number
85
of poles and to maximize the information content of the U matrices. As a result, in
the Krylov basis, one has to diagonalize potentially both large and ill-conditioned
matrices, which is not feasible in practice. For example, a normal 2D NMR signal
with 1024 × 256 complex data points will lead to U matrices of dimension 65536 by
65536, which can hardly be handled even by modern supercomputers.
Instead of trying the solve the 2D HIP in the full space, a more efficient approach
is to break it into many small problems in the subspace. This can be achieved by
using a 2D Fourier-type localized basis, among which the rectangular-window Fourier
basis6 is particularly efficient [68],
|Ψj) =M1−1∑
n1=0
M2−1∑
n2=0
ei~n(~ϕj−~Ω)|Φ0) , (2.50)
where ~ϕj ≡ (ϕ1j, ϕ2j), j = 1, 2, . . . , Kwin, is a 2D frequency grid within a small pre-
specified 2D spectral window [ωmin1, ωmax1] × [ωmin2, ωmax2]. The 2D Fourier basis is
locally complete, so that small data matrices can be constructed and then diago-
nalized to obtain accurate spectral estimations within the local spectral window. In
principle, any type of 2D frequency grid can be used as long as the basis distribution
is sufficiently uniform and the total number of basis functions Kwin satisfies,
Kwin = ρ∆ω1τ1 × ∆ω2τ2
4π2M1M2 , (2.51)
with ρ ≥ 1.0 being the density of basis functions, and ∆ωl = ωmaxl − ωminl, l =
1, 2. A particularly efficient setup corresponds to having a uniform direct-product
2D grids: ~ϕj = (ϕj1, ϕj2), j1 = 1, 2, . . . , Kwin1, j2 = 1, 2, . . . , Kwin2, where Kwin1 =
6In principle, a 2D multi-scale Fourier basis can be used [79]. However, as it will become clearlater, there are other problems in MD-FDM, making the necessity of using such a sophisticated localbasis less obvious.
86
ρ1∆ω1τ1M1/2π and Kwin2 = ρ2∆ω2τ2M2/2π. Thus total basis size Kwin = Kwin1 ×
Kwin2, and overall basis density ρ = ρ1ρ2. The U matrix elements in such a 2D
Fourier basis can be efficiently computed using a procedure similar to the 1D case
(see Section 2.2). The final expressions in a compact form can be directly deduced
from Eq. 2.21 and Eq. 2.23. They are given here7,
[
U(~p)]
jj′= S1S2
∑
σ1=0,1
eiσ1[M1τ1(ϕ1j′−ϕ1j)+π]
1 − eiτ1(ϕ1j′−ϕ1j)
(σ1+1)(M1−1)∑
n1=σ1M1
(2.52)
×∑
σ2=0,1
eiσ2[M2τ2(ϕ2j′−ϕ2j)+π]
1 − eiτ2(ϕ2j′−ϕ2j)
(σ2+1)(M2−1)∑
n2=σ2M2
ei~n~ϕjc(~n + ~p) ,
where Sl defines the symmetrization operator over the subscripts lj and lj′ as in
Eq. 2.22. When ϕlj = ϕlj′ Eq. 2.52 is rewritten according to Eq. 2.23. For example,
for ϕ1j = ϕ1j′ we have
[
U(~p)]
jj′= S2
∑
σ2=0,1
eiσ2M2τ2(ϕ2j′−ϕ2j)+π]
1 − eiτ2(ϕ2j′−ϕ2j)
(σ2+1)(M2−1)∑
n2=σ2M2
(2.53)
×2(M1−1)∑
n1=0
(M1 − |M1 − 1 − n1|) ei~n~ϕjc(~n + ~p)
which can be trivially rewritten for the symmetric case of ϕ2j = ϕ2j′. For the case
~ϕj = ~ϕj′,
[
U(~p)]
jj=
2(M1−1)∑
n1=0
2(M2−1)∑
n2=0
ei~n~ϕjc(~n + ~p) (2.54)
× (M1 − |M1 − 1 − n1|)(M2 − |M2 − 1 − n2|) .
Once the U matrices are computed, one then needs to solve the 2D generalization
of the 1D generalized eigenvalue problem,
UlBk = ulkU0Bk , l = 1, 2 . (2.55)
7Similar to the 1D case, tilde ∼ is added the notations B, U and others to indicate the use ofthe Fourier basis.
87
The amplitudes dk are then obtained from the eigenvectors as
d1/2k = B
T
k C , (2.56)
where the coefficients of the 1×Kwin column vector C are computed using the following
2D FT of the original signal array c(~n):
[
C]
j=
M1−1∑
n1=0
M2−1∑
n2=0
ei~n~ϕjc(~n) , j = 1, 2, ..., Kwin . (2.57)
After the parameter list ω1k, ω2k, dk is computed, 2D spectral estimations can be
constructed. The 2D complex spectrum that corresponds to infinite time 2D discrete
Fourier summation,
I(ω1, ω2) = τ1τ2
∞∑
n1=0
∞∑
n2=0
ein1τ1ω1ein2τ2ω2
(
1 − δn10
2
)(
1 − δn20
2
)
c(n1τ1, n2τ2) , (2.58)
can be obtained by substituting Eq. 2.42 into Eq. 2.58 and evaluating the summations
analytically:
I(ω1, ω2) = τ1τ2
∑
k
dk
[
1
1 − eiτ1(ω1−ω1k)− 1
2
] [
1
1 − eiτ2(ω2−ω2k)− 1
2
]
. (2.59)
In many applications, a double absorption spectrum is desirable. 2D FT of purely
phase modulated signals gives rise to “phase-twisted” lineshape [88]. Neither the real
nor the imaginary part of I(ω1, ω2) yields double absorption line shape. A double
absorption spectrum can only be obtained by using both cosine- and sine-modulated
or N- and P-type data sets (see Section 1.2.3). However, in FDM, the parametric
representation ω1k, ω2k, dk is computed using a single purely phase-modulated data
set, thus a double absorption type of spectrum can be constructed, for example, using,
A(ω1, ω2) ≈ τ1τ2
∑
k
Re[dk] Re[
1
1 − eiτ1(ω1−ω1k)− 1
2
]
Re[
1
1 − eiτ2(ω2−ω2k)− 1
2
]
,
(2.60)
88
where we assume that the signal is correctly phased. However, this expression is
not a unique representation, and is meaningful only when the different peaks with
absorption shape are not overlapping too much, i.e., the interference effects are not
significant. Alternatively, one can use use an expression where the interference effects
in the second dimension are conserved,
A(ω1, ω2) ≈ τ1τ2
∑
k
Re[
1
1 − eiτ1(ω1−ω1k)− 1
2
]
Re[
dk
(
1
1 − eiτ2(ω2−ω2k)− 1
2
)]
. (2.61)
Equations 2.52 through 2.57 outline an efficient way of solving the 2D HIP of
Eq. 2.42 using pure linear algebra in the frequency subspace. The two most important
properties of 1D FDM are preserved. Therefore, 2D FDM is also intrinsically superior
in efficiency and numerical stability. However, there is a new problem. The essential
step of 2D FDM is solving the 2D generalized eigenvalue problem of Eq. 2.55, which
requires finding a unique set of eigenvectors that simultaneously diagonalizes both
evolution matrices. As there is no existing 2D generalized eigenvalue problem solver,
in practice 1D generalized eigenvalue solver is used twice to diagonalize U1 and U2
separately,
UlBlk = ulkU0Blk , l = 1, 2 . (2.62)
Two sets of eigenvectors Blk, l = 1, 2 are obtained and then need to be matched.
The matching step is typically problematic due to several factors: (i) the presence
of noise and other imperfections; (ii) use of the Fourier subspace; (iii) degeneracy of
eigenvalues. As a result, even though the Hamiltonians Ω1 and Ω2 are assumed to
commute with each other, the U matrices constructed in the Fourier basis for noisy
signals typically do not commute anymore. Therefore, an unique set of eigenvectors
89
does not necessarily exist. There are various attempts at trying to find some unique set
of eigenvectors by reformulating the 2D generalized eigenvalue problems [66, 68, 69] or
by “simultaneous diagonalization” [70]. However, they are only applicable to signals
with very high SNR and fail for realistic signals.
Nevertheless, applied to model signals, even such a “naive” version of FDM al-
ready shows the power of true multi-dimensional data processing. Figure 2.12 is an
interesting demonstration of 2D FDM using a nearly noiseless model signal. The
signal consists of 32 non-degenerate 2D Lorentzian lines with frequency coordinates
created by a random number generator. The average line width in both dimension is
0.02 Hz and all the peaks have the same integral. The spectral width is 2 Hz in both
dimensions. The complete data set contains 128 × 128 complex data points. In the
2D DFT calculations, cosine weighting function was always used in both dimensions.
Double absorption FT spectra were obtained using both N-type and P-type signals.
The FDM calculation only used the N-type signal as double-absorption type of spec-
tra can be computed from a single purely phase-modulated signal. The left column
shows the 2D DFT spectra obtained by using different number of data points. We can
clearly see the effects of the FT uncertainty principle. Whenever the signal is short in
a time dimensions, the FT resolution is low in the corresponding frequency dimension
(panel a and b). It is only when the signal is long in both time dimensions that FT
can obtain high resolution in both frequency dimensions (panel c). The 2D FDM
spectra shown in the right column tell a totally different story. As long as the total
size of the signal is sufficiently large, whether it is 32 × 8 (panel d) or 8 × 32 (panel
e) has little effect on the final results: FDM can always provide a fully resolved 2D
90
Ntot = 32x8
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Ntot = 32x8
Ntot = 8x32Ntot = 8x32
Ntot =128x2Ntot = 64x64
(Hz) (Hz)
2D FT 2D FDM
a)
b)
c)
d)
e)
f)
Figure 2.12: A demonstration of FDM using a nearly noiseless model signal that con-tains 32 2D Lorentzian lines. 2D DFT can provide high resolution in both dimensionsonly when the signal is long both ways. However, for FDM, as long as the area of the2D signal is sufficiently large, fully resolved spectra can always be obtained, even inthe extreme case where the signal has only 2 increments in one dimension.
91
spectrum. This is exactly what we expect from a true 2D spectral analysis method, as
illustrated in Figure 2.11. Even in the extreme case where there are 128 data points
in the first dimension and only 2 increments in the second dimension (panel f), FDM
still fully resolves all 32 peaks, since the total size is sufficiently large.
Computational Cost of 2D FDM
There are two steps that are time consuming in 2D FDM: construction of the U
matrices and diagonalization of these matrices. By using a direct product frequency
grid, one can use the efficient expressions of Eqs. 2.52, 2.53 and 2.54. The computa-
tional cost of the double summations in Eq. 2.52 scales as N1 ×N2. We only need to
evaluate these double summations once for each of Kwin basis functions. Accordingly,
the total cost of computing the U matrices scales as Kwin × N1 × N2. The cost of
diagonalization step is of the order of K3win. Thus the total computational cost of
each FDM single window calculation is Kwin(a K2win + bN1N2), with a and b being
scaling factors (a > b in our case).
To analyze the whole spectral range, one needs to carry out the order of N1N2/Kwin
window calculations, leading to a total cost of N1N2(a K2win + bN1N2). In the limit
of small windows, where K2win N1N2, the total numerical effort of analyzing the
whole spectral range using FDM scales as N21 × N2
2 , which is of the same order as
that of 2D DFT. In the case of short signals, where K2win is comparable to or greater
than N1 × N2, the computational cost will have a strong dependence on the window
size. For example, to compute the 2D spectral estimation over the whole spectral
range for a 2D signal with 1024×256 complex data point using basis size Kwin = 400,
92
the total processing time is about 10 hours on a small AMD XP 1800+ CPU Linux
workstation. In most cases, only a small portion of the spectral range is of interest.
Therefore the actual processing time is often less than one hour for most 2D NMR
applications.
2.3.4 Green’s Function Approach
When the signal is noisy and/or contains degenerate poles, the “naive” version of
2D FDM simply fails due to the difficulty of finding an unique set of eigenvectors that
simultaneously diagonalizes both Hamiltonians. In this subsection, we are going to
show that there is a way to avoid this problem and construct 2D spectral estimations
directly from the results of two 1D generalized eigenvalue problems in terms of Green’s
functions [71]. A new derivation that is very similar to that of the Regularized
Resolvent Transform [77] is given here.
First, let’s redefine the object of 2D FDM. Given a finite 2D signal c(~n) ≡
c(n1τ1, n2τ2) defined on an equidistant 2D time grid, instead of trying to obtain a
compact parametric representation ω1k, ω2k, dk, the new object is to estimate the
infinite time 2D DFT spectrum, defined in Eq. 2.58, using the finite signal. Inserting
the quantum ansatz of Eq. 2.43 into Eq. 2.58 and evaluating the geometric summa-
tions analytically gives
I(ω1, ω2) =(
Φ0
∣
∣
∣ G1(ω1) G2(ω2)∣
∣
∣Φ0
)
(2.63)
=
(
Φ0
∣
∣
∣
∣
∣
1
1 − eiτ1ω1U1
− 1
2
1
1 − eiτ2ω2U2
− 1
2
∣
∣
∣
∣
∣
Φ0
)
,
where Gl are resolvent operators given in terms of Green’s functions. Substituting
93
the spectral representation of the evolution operators,
Ul =∑
k
ulk |Υlk) (Υlk| , l = 1, 2 , (2.64)
into Eq. 2.63, we can obtain an expression for computing the 2D complex spectrum
directly from the eigenvectors and eigenvalues,
I(ω1, ω2) = τ1τ2
∑
k,k′
Dk,k′
[
1
1 − eiτ1(ω1−ω1k)− 1
2
] [
1
1 − eiτ2(ω2−ω2k′)− 1
2
]
, (2.65)
with cross-amplitudes Dk,k′ defined as,
Dk,k′ = b1kTk,k′b2k′ = (Φ0|Υ1k)(Υ1k|Υ2k′)(Υ2k′|Φ0) . (2.66)
In addition, one can easily calculate the 1D projections in both dimensions,
Il(ωl) =(
Φ0
∣
∣
∣ Gl(ωl) Φ0
)
=∑
lk
b2lk
[
1
1 − eiτ2(ω2−ω2k′ )− 1
2
]
, (2.67)
where blk ≡ (Υlk|Φ0) as defined in Eq. 2.66. Finally, a double-absorption type of
spectral estimation can be computed as,
A(ω1, ω2) ≈ τ1τ2
∑
k,k′
Re[Dk,k′]Re[
1
1 − eiτ1(ω1−ω1k)− 1
2
]
Re[
1
1 − eiτ2(ω2−ω2k′ )− 1
2
]
,
(2.68)
where we assume that the signal is correctly phased and that different peaks with
absorption shape are not overlapping too much so that the interference effects are
not significant.
Evaluated in the 2D Fourier basis, defined in Eq. 2.50, the operator expressions
of Eqs. 2.65, 2.67 and 2.68 become working formula for computing corresponding
spectral estimations with,
blk = BT
lkC , (2.69)
Tk,k′ = BT
1kU0B2k′ , (2.70)
94
where the vector C is defined in Eq. 2.57, and eigenvectors B are obtained from
solving the two 1D GEPs of Eq. 2.62.
Note that cross matrix Tk,k′ = (Υ1k|Υ2k′) is not necessarily diagonal; nor can
it necessarily be reduced to diagonal form by permutations. However, if we let
(Υ1k|Υ2k′) = δk,k′, Eq. 2.65 boils down to the ideal representation of Eq. 2.59. Thus
adopting representation of Eq. 2.65 covers the ideal case when two evolution opera-
tors can be simultaneously diagonalized, and at the same time avoids the necessity
of generating a unique set of eigenvectors Υk. As a result, it more robust and
applicable to processing general noisy experimental signals.
2.3.5 Averaging Approaches to Suppress the Artifacts
In principle, most entries in Tk,k′ should be nearly zero for signals that give rise
to spectra without excessive direct-product patterns. Unfortunately, in practice, this
is not necessary the case, which manifest itself as the spurious spikes in the 2D
spectral estimations, even for the signals with a reasonable signal to noise ratio.
For example, Figure 2.13 shows several double absorption type of 2D FT and FDM
spectra, computed from a 1H-15N chemical shift correlation NMR signal of 15N labeled
metalloprotein rubredoxin [89]. Only 8 increments have been used in 15N dimension
with 300 complex data points along the proton dimension. Even though the FDM
spectra seem to show some high-resolution characters, they are contaminated by
various artifacts such as spurious spikes randomly distributed all over the spectrum
and poorly converged genuine poles. Moreover, the artifacts are very sensitive to both
95
-800
-600
-400
-200
0
200
400
600
800
-600-400-2000200400600800 -600-400-2000200400600800
-800
-600
-400
-200
0
200
400
600
800
2D FT 2D FDM
2D FDM Multi-Scale 2D FDMwith different parameters
1H / Hz 1H / Hz
15N
/ Hz
15N
/ Hz
Figure 2.13: 2D FDM and 2D FT double absorption spectra of a 1H-15N chemical shiftcorrelation 2D NMR experiment of a 15N labeled metalloprotein, rubredoxin. The 2Dtime signal contains 300×8 complex points. While showing some high-resolution fea-tures, the spectra obtained from a single FDM calculation are contaminated withartifacts such as various spurious spikes randomly distributed throughout the wholespectral window and lineshape distortions of genuine peaks. These artifacts are highlysensitive to any change of the input signal and FDM parameters (e.g, compare thetwo 2D FDM spectra computed using same data set but with slightly different pa-rameters). Using a Multi-Scale Fourier basis does not help to suppress the artifacts(lower right panel).
96
the small variations in the input data and the parameters of the FDM calculation.
Multi-Scale Fourier basis does not help to suppress these artifacts (see lower right
panel), indicating that this is a different kind of instability. One might suspect that
this is due to the fact that the eigenvectors are computed in a non-variational way
and therefore less accurate than the eigenvalues. However, there are more essential
reasons for it. How to improve the structure of Tk,k′ is one of most important problems
in 2D FDM. These will be discussed in details next section.
Despite the complexity of the problem, there are several straightforward averaging
procedures for suppressing the artifacts, which simply make use of their high sensitiv-
ity to any change of input data and FDM processing parameters. These procedures
were successfully used in early applications of multi-dimensional FDM [69, 72, 73, 90].
The first averaging procedure is signal-length averaging. It is based on multi-
ple applications of FDM, applied to the nested subsets of the same signal, using a
progressively larger total size in successive calculations. As most artifacts are very
sensitive to any change of FDM parameters (while the true features are more stable),
they can be averaged out by summing many ersatz spectra computed from different
sizes of subsets of the signal. In principle, any FDM parameters such as basis size,
basis density and window positions can be used in the same context. However, it was
found that varying the signal size is particularly robust. In addition, typical 2D NMR
signals have many data points in acquisition dimension, providing a large range for
changing the signal size to achieve sufficient averaging. Examples of the signal-length
averaging procedure can be found in Ref. [69, 72].
The second procedure, pseudo-noise averaging, is a little more elegant and does
97
not require a long dimension to provide sufficiently large room for varying the signal
length. In stead, it exploits the great sensitivity of FDM ersatz spectrum to small
perturbations of the time signal. A small change δ(~n) in the time signal call lead
to a large variation of the ersatz spectrum. Similar to the case of varying the signal
length, the artifacts are more sensitive than the true features. Therefore, by averaging
over sufficiently many realizations of pseudo-noise perturbation one can suppress the
artifacts in the ersatz spectrum. Interestingly, the pseudo-noise perturbation can be
implemented right before solving the generalized eigenvalue problems of Eq. 2.62: by
adding pseudo-random noise to the U matrices computed from the original signal,
one can save the computation time spent on re-computing the U matrices for each
perturbation. The qualitative justification is that U matrices are linear functions of
the input time signal and therefore a variation of the time signal transfers linearly
into the variation of the U matrices [91]:
Ul → Ul + rY , (2.71)
where Y is a complex symmetric matrix with independent random elements satisfying
〈[Y]jj′〉 = 0, 〈|[Y]jj′|2〉 = 1. r is a real number and defines the level of perturbation,
which, in practice, can be calculated as,
r =q
K2win
Kwin∑
j=1
Kwin∑
j′=1
∣
∣
∣[Ul]j,j′∣
∣
∣ , (2.72)
where q is a real number. Typical value of q ranges from 0.01 to 0.05 depending
on how persistent’ the artifacts are with respect to the perturbations. 20 to 100
perturbations are usually sufficient to provide stable averaged 2D spectra that are
free from most artifacts. Figure 2.14 shows a stable, clean and high-resolution 2D
98
double absorption spectrum obtained by FDM pseudo-noise averaging, applied to the
same signal used to in Figure 2.13.
-800-600-400-2000200400600800
-800
-600
-400
-200
0
200
400
600
800
1H / Hz
15N
/ Hz
Figure 2.14: 2D FDM double absorption spectrum computed from the same signalused in Fig.2.13. Only 300×8 complex points were used for the purpose of demonstrat-ing the high-resolution nature of FDM. Results of 50 FDM calculations were co-addedand averaged with the pseudo-noise perturbation level of q = 0.02 (see text). Thefinal spectrum is well resolved (see Fig. 2.13 and Fig.2.17 for 2D FT spectra using300 × 8 and 300 × 128 data points respectively), and free of most artifacts.
An efficient implementation of the pseudo-noise averaging was given in Ref. [91].
The expression for computing a 2D double-absorption spectral estimation with pseudo-
noise averaging is given here. We simply rewrite Eq. 2.68 as
Aτ (ω1, ω2) = X1q(ω1)TU0X2q(ω2) , (2.73)
99
where the absorption-mode vectors Xlq(ωl) are computed independently for each
l = 1, 2 using the eigenfrequencies and eigenvectors of the corresponding perturbed
generalized eigenvalue problems,
(
Ul + rY)
Blk = ulkU0Blk , (2.74)
and then averaged over a sufficient number, NFDM, of realizations of the random
perturbation Y as
Xlq(ωl) = τl
⟨
∑
k
Re[
1
1 − eiτl(ωl−ωlk)− 1
2
]
BlkBT
lkC
⟩
Y
. (2.75)
The obvious drawback of both averaging procedures is that the computational
cost is multiplied the number of FDM averaging, NFDM. The result improves roughly
as√
NFDM, assuming the artifacts are completely random for different calculations.
It was realized later that the averaging procedure was actually a general but primi-
tive realization of regularization for solving ill-conditioned problems. More efficient
regularization techniques such as Singular Value Decomposition (SVD) [9, 92] and
Tikhonov Regularization [93] were then introduced to FDM, where only a single
FDM calculation is required for obtaining stable and clean 2D spectral estimations.
They will be discussed in the next section.
2.3.6 Summary
The 1D FDM algorithm was successfully extended to processing multi-dimensional
signals. The two most important properties of 1D FDM, namely, solving the non-
linear fitting problem by pure linear algebra and local spectral analysis by using a
100
Fourier-type localized basis, both apply to MD-FDM. Thus MD-FDM is intrinsically
efficient and stable method for fitting the time signal to the summation of multi-
dimensional Lorentzian lines.
One of the most interesting properties of MD-FDM is that it is a true multi-
dimensional spectral analysis method, where the obtainable resolution in all dimen-
sions is determined together by the total information content of the signal. MD-FDM
is able to process the whole data set to characterize the intrinsically multi-dimensional
features and provide high-resolution in all dimensions, given that the signal sufficiently
satisfies the Lorentzian model and is sufficiently long.
There are also some new problems associated with processing multi-dimensional
time signals. Straightforward MD extensions of FDM require finding a unique set of
eigenvectors to simultaneously diagonalize multiple data matrices. Such a solution
might not exist at all in the cases of noisy signals with potential degeneracies. It can
be avoided by using a Green’s function approach, in which 2D spectral estimations
can be directly computed from the results of solving several 1D generalized eigenvalue
problems. With Green’s function formalism, FDM can be applied to analyzing general
noisy time signals. However a meaningful, compact parametric representation is
typically no longer available.
Finally, spectra obtained by a single FDM calculation are still contaminated with
artifacts which are very sensitive to any change of input signal and FDM processing
parameters. This problem is very different from what we have seen in 1D case, and
can not be solved by using a multi-scale Fourier basis. Some primitive averaging
procedures that simply make use of high sensitivity are introduced to suppress the
101
artifacts. While stable high-resolution spectra can be obtained, the averaging signifi-
cantly increases the computational cost of MD-FDM and therefore is not the optimal
solution. More efficient methods for suppressing the artifacts will be discussed in the
next section.
2.4 Regularization of Multi-Dimensional FDM
In this section, we will further address the problem of ill-conditioning in multi-
dimensional FDM. The averaging procedures we discussed in the previous section are
effective but computationally expensive and converge slowly. Further study of the
problems helps us to understand the sources of the instability better and makes it
possible for us to develop more efficient methods of regularizing FDM. Again, we use
2D FDM as an example to illustrate the principles.
2.4.1 Ill-conditioning of 2D FDM
The source of the instability can be traced back to the 2D HIP. In the 1D case,
given N = 2M complex data points in the time domain, there exists an exact and
unique set of M pairs ωk, dk that satisfies the 1D HIP equation Eq. 2.1, if there
is a solution8, even though there is no guarantee that such a fit is a “good” fit in
the sense of all poles being physical and truly present in the time signal. However,
the 2D HIP is ill-posed. Given a 2D signal of the size of N1 × N2, the maximum
number of unknowns that we can obtain from solving the 2D HIP of Eq. 2.42 is only
8In the case of degeneracy, there might be no solution to the 1D HIP.
102
3K = 3M1M2 = 3N1N2/4 < N1 × N2. Therefore, the 2D HIP is posed as an over-
determined problem9. For a general 2D signal, there is no exact solution unless the
Lorentzian assumption is strictly satisfied and the size the signal is sufficiently large
to recover all the poles. In practice, we almost always end up with ill-conditioned
generalized eigenvalue problems due to the following reasons:
i) Realistic signals always contain noise which does not satisfy the 2D Lorentzian
model. In particular, multi-dimensional NMR signals are especially noisy due to lower
sensitivity, t1 noise, and limited long term instrumental stability.
ii) The peaks do not have perfect Lorentzian lineshapes due to many experimen-
tal limitations, such as imperfect decoupling, long range couplings, inhomogeneity
of magnetic field and many other factors. However, in some special cases such as
Constant-Time experiments [75, 76], it is possible to enforce the Lorentzian lineshape
in the indirect dimensions and thus improve the performance of FDM significantly.
iii) A Fourier-type localized basis has to be used to avoid the problem of solving
huge and ill-conditioned systems. However, the Fourier filter is not perfect. Interfer-
ences from nearby features and nonlocalized features (such as very broad poles that
define the baseline) act effectively as some additional “noise” to the local spectral
analysis. One might try to use a more sophisticated basis such as a 2D Multi-Scale
Fourier basis to minimize the window effects. However, numerical experiments seem
to suggest that the 2D multi-scale Fourier basis does not make a significant differ-
ence. This could indicate that 2D Mulit-Scale Fourier basis is not as efficient as the
9The 2D HIP can be also formulated using direct product poles ω1k, ω2k′ , dkk′ with 2K +K2 =N1N2/2 + N2
1N2
2/16 unknowns. It is then posed as an under-determined problem except for the
case of very small signals (N1N2 ≤ 8), which in general, has infinite number of solutions and arethus very unstable numerically.
103
1D version and that there are still other limiting factors.
iv) The total size of the 2D signal is typically much greater than the total number
of peaks present. For example, let’s consider a 1H-13N chemical shift correlation
experiment of a medium-size protein of 200 residues. A typical signal could contain
1024× 256 data points or even more, from which a maximum number of 512× 128 =
65536 peaks could be determined, much greater than the number of possible peaks,
which is of the order of a few hundred. Therefore, in practice, the 2D HIP problem
is even more over-determined.
As a result, the data matrices that occur in 2D FDM are almost always nearly
singular. The condition numbers10 of the U matrices are typically greater than 1010
in 2D FDM, while in 1D FDM the condition number rarely exceeds 105. Exact solu-
tion to such an ill-conditioned problem is very unstable and often meaningless. This
is exactly what we have seen previously in Figure 2.13. So called regularization tech-
niques are required in order to obtain a stable and meaningful solution. Simply put,
regularization is the numerical procedure of removing the ill-conditioning of the prob-
lem. For example, regularization can be implemented by adding some “smoothness”
constraint [94],
||R X − C|| + q ||X|| = min. , (2.76)
with q being the regularization parameter, when solving an ill-posed linear system
R X = C. However, unlike the case of solving a linear system, it is not obvious how
the conventional regularization techniques, such as SVD and Tikhonov regularization,
10The condition number is a numerical measurement of the conditioning of a matrix. It can bedefined as the ratio of largest and smallest singular values of the matrix.
104
can be applied to the generalized eigenvalue problems encountered in FDM. Further
complications include the lack of 2D generalized eigenvalue solvers and the difficulty
of finding a unique set of eigenvectors that simultaneously diagonalizes both evolution
matrices. We have to use the Green’s function approach, where two 1D generalized
eigenvalue problems are solved independently. It is not clear how we can regularize
both eigenvalue problems in a systematic, consistent fashion. Further more, as the
eigenvectors are obtained in a nonvariational way for typical eigenvalue solvers, the
numerical errors in each set of the eigenvectors are larger (compared to the eigen-
values), which is another source of many non-zero entries in the cross matrix Tk,k′,
defined in Eq. 2.69.
In the rest of this section, we will discuss three regularization methods that have
been applied to 2D FDM. Of these, the FDM2K algorithm [74] is particularly efficient
and is used currently as the method of regularization for FDM.
2.4.2 Singular Value Decomposition
Singular Value Decomposition, or, SVD, is a very powerful technique for diagnos-
ing the matrices that are either singular or numerically close to singular. In certain
cases, SVD can also help to solve the problem by isolating the sources of instability,
even though one should be careful in analyzing the “answer” from SVD. Any N ×N
square11 matrix R can be decomposed into the product of three unique matrices,
R = WΛV†, (2.77)
11Note that SVD can also be applied to a rectangular matrix.
105
where W and V are N × N unitary matrices, and Λ = diagλi, a real diagonal
matrix with λ1 ≥ λ2 ≥ ... ≥ λN ≥ 0 being singular values. The superscript † denotes
“conjugate transpose”. The columns of matrix W, or equivalently, those of matrix
V, form an orthonormal set of basis vectors which covers both the nullspace12 (if any)
and range of matrix R. All the singularity or ill-conditioning of R is contained in the
diagonal matrix Λ. By examining the singular values, it is possible to determine the
rank and separate the nullspace of R. Specifically, the columns of W, whose same-
numbered singular values λi are nonzero, form an orthonormal set of basis vectors
that span the range; the columns of V, whose same-numbered singular values λi are
zero, form an orthonormal basis for the nullspace [9]. Unfortunately, reals situations
are more complicated than this ideal assumption. SVD of the U matrices computed
from a real signal with noise will contain only nonzero singular values. There are
only ’large’ and ’small’ singular values. One can then hope that plotting λj vs. j
shows some abrupt magnitude change so that a zero threshold can be unambiguously
picked. This might be true in some cases but often not in NMR applications, making
choosing a zero threshold problematic.
There are several ways SVD can be used to regularize FDM. They can be divided
into two categories: hard and soft regularization.
1. Hard Regularization: truncated SVD.
In this implementation, we apply SVD to U0 and assume that it is possible to
determine the rank by choosing some zero threshold. Once the rank is identified, we
12For a singular matrix R, there exists some vectors x in some subspace, called nullspace, thatsatisfy R · x = 0. The dimension of the nullspace, defined by the number of linearly independentvectors x, is called the nullity of R.
106
can re-evaluate the U matrices in the reduced basis formed by columns of W that
correspond to the singular values greater than the zero threshold,
[Ured
l ]ij = WTi UlWj . (2.78)
The column vector C defined in Eq. 2.57 should be also re-evaluated in the new basis,
[Cred
]i = WTi C . (2.79)
Note that SVD of U(1)
and U(2)
should give similar results. However, U0 is the
least noisy and thus preferred. Also note that the transpose instead of the conjugate
transpose of Wi is used, as the complex inner product instead of the Hermitian inner
product is always used in FDM for describing dissipative systems.
As pointed out by Moler and Stewart [80], even when the matrices on both sides
of the GEP have a common null space (as is the case in FDM), it is not recommended
to use SVD to get rid of the null subspace. The reason is that the eigenvalues and
eigenvectors become very sensitive to the assumed rank of range subspace. On the
other hand, it was argued that the QZ algorithm [80] provided accurate eigenvalues
and eigenvectors in terms of two numbers, uk = αk/βk, the accuracy of which was
not affected by ill-conditioning of the matrices. It was also argued that “unreliable”
eigenvalues could be identified by smallness of both αk and βk. Our experience with
2D FDM is somewhat contradictory to these recommendations. Spectra constructed
from the accurate eigenvalues and eigenvectors computed by QZ, applied to the orig-
inal U matrices, are contaminated by artifacts, while the truncated SVD of U0 can
help in removing the artifacts, as demonstrated in Figure 2.15. Despite this, the
ambiguity in deciding how many basis vectors to retain remains a major problem.
107
Figure 2.15 shows four 2D 15N − 1H chemical shift correlation spectra obtained
from a sample of 15N labeled rubredoxin, a small metalloprotein [89]. Only 16 incre-
ments are used in the 15N dimension, with 200 complex points along the 1H dimension.
The initial basis size is then 50 × 8 = 400. Plotting the singular values (top panel)
clearly shows an abrupt magnitude change at around i ∼ 30. Setting a cutoff at this
point will give a very clean, well converged spectrum, panel (c). We do not actually
need to throw away so many basis vectors. Panel (b) shows that even throwing away
only 23 basis vectors corresponding to the smallest 23 singular values already reduces
most of the artifacts. Further reduction of the basis size to around 150 will basically
yield the same spectrum as shown in panel (c) (data not shown). While it is safer to
keep a few more basis vectors,it is extremely dangerous to throw away more vectors
than necessary, which could lead to missing genuine peaks, e.g., panel (d). This is a
big disadvantage of such an aggressive procedure. Even worse, typical NMR data may
not have such a clean break in singular values. Then choosing any cutoff can be very
risky and give the operator a chance to bias the experimental results to support some
particular viewpoints. Therefore, such a procedure is not generally applicable. It is
better to be more conservative, applying “soft cutoff” instead of the “hard cutoff”.
2. Soft Regularization: Pseudo-Inverse of U0: 1/λj → λj/(λ2j + q2)
In this implementation, we solve the GEP of equation Eq. 2.62 in two steps: first
we apply SVD to U0 and compute a pseudo inverse,
U−1
0 (q) = V [diag(1/λj)]q W†, (2.80)
which is dependent on a real regularization parameter q. Then, we solve the normal
108
0 30 600.0
0.5
1.0
0 100 200 300 4000.0
0.5
1.0
(Index i)
(Nor
mal
ized
Sin
gula
r V
alue
s λ i/λ
max
)
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0
200
400
600
a. Keep all 400 basis vectors
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200
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0
200
400
600
c. Keep 28 basis vectors
-600-400-2000200400600
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-400
-200
0
200
400
600
d. Keep 14 basis vectors
Figure 2.15: Chemical shift correlation spectra obtained by FDM with hard reg-ularization via truncated SVD (see text). The total size of the signal used isN1 × N2 = 200 × 16, leading to a total basis size is Nbtotal = 50 × 8 = 400 (asingle window was used). The top panel shows the distribution of singular valuesof the overlap matrix U0. The four spectra (a,b,c,d) at the bottom are obtained byretaining various numbers of basis vectors.
109
eigenvalue problems,
U−1
0 (q) Ul Blk = ulkBlk l = 1, 2 . (2.81)
There are several possible ways of evaluating the pseudo-inverse matrix U−1
0 (q).
One can simply use the conventional truncated SVD fashion pseudo inverse: replace
all the 1/λj by zero if λj is smaller than some zero threshold q. Then the results
will be also sensitive to the choice of zero threshold just like the hard regularization.
A less aggressive way is to replace all 1/λj with λj/(λ2j + q2) with q being a real
regularization parameter. The advantage of this approach is that the regularization
is “softer” and small deviation of q from the optimal value will not lead to dramatic
deterioration of the results. There is usually a stable region where changing q2 up to
one order of magnitude leads to little change in the 2D spectral estimation, provided
that the signal contains enough information for FDM to obtain a stable estimation.
In practice, the optimal level of regularization can by found by carrying out multiple
calculations with different values of q2.
Shown in Figure 2.16 are several 2D double absorption spectra obtained by single
FDM calculations with different levels of regularization, applied to the same signal
as that used in Figure 2.15. When the regularization is too small, the spectra is still
contaminated with artifacts, e.g., panel (a); further increasing the regularization to
a sufficiently large level results in a stable and high-resolution 2D spectra, shown in
panel (b). There is a region where the spectra obtained are relatively stable according
to the change of q. Finally, when the regularization is too large, the FDM spectrum
is furthered smoothed and small features are suppressed down to the baseline, e.g.,
110
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400
200
0
-200
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600
400
200
0
-200
-400
-600
-800
b. q = 0.01
d. q = 0.1
1H / Hz 1H / Hz
15N
/ Hz
15N
/ Hz
Figure 2.16: Chemical shift correlation spectra obtained by FDM using soft regu-larization via SVD (see text). The signal used is the same as that used in Figure2.15. The signal length processed is N1 × N2 = 200 × 16 and total basis size isNbtotal = 50 × 8 = 400. It shows that with appropriate level of regularization, stableand high-resolution spectra can be obtained using single FDM calculation (panel b:q = 0.01). In addition, the appearance of the spectra has a smooth dependence onthe regularization parameter (panel b, c and d). When the regularization is too large,small features are suppressed to baseline (panel d). Note that the overall spectrumis smoothed in a very non-uniform way. The values of q shown are ratios of real q tothe largest singular value (λ1).
111
panel (d). Moreover, the spectrum is smoothed in a non-uniform way. We can
pretend to understand the effect of the regularization simply by saying that increasing
q de-emphasizes the contribution of small singular values, which are responsible for
describing the noise, small features and fine structures. However, exactly how a
particular peak might be affected by the regularization is not totally clear.
2.4.3 FDM2K
The soft regularization via pseudo inverse of U0 is a very efficient method in
suppressing the artifacts. However, SVD is expensive and therefore suboptimal com-
putationally. There is another very effective but cheaper way of regularizing FDM,
which was invented in the year of 2000 and therefore named FDM2K. In FDM2K, we
simply rewrite the original GEP of Eq. 2.62 as,
U†0UlBlk = ulk
(
U†0U0 + q2
)
Blk , (2.82)
by introducing a real, positive regularization parameter q. The new generalized eigen-
value problem now has a positive definite Hermitian matrix on the right hand side.
It can be proven that q imposes minimum singular value for the new right matrix
and therefore effectively controls its condition number. Similar to the case of soft
regularization, the resulting spectra will have a relatively smooth dependence on the
regularization level. Figure 2.17 demonstrate the FDM2K algorithm using the same
15N-1H HSQC signal of rubredoxin used in previous examples. When signals are suf-
ficiently long and have reasonably high SNR13, there will be a stable region where the
13This qualitative statement will become more quantitative in Chapter3, where we will study thedependence of performance of FDM2K on SNR and signal sizes.
112
spectra obtained change very little while the regularization changes by up to an order
of magnitude. When the regularization is too high, small features are smoothed out
first while the strong peaks are almost unaffected, and the overall smoothing effect is
non-uniform.
It is very tempting to treat q as the “noise power”, just like in the Maximum
Entropy applications [8]. Indeed, it can be proved semiquantitatively [74], that the
spectral features with amplitudes dk of order of q and below are smoothed out, while
the stronger peaks remain essentially unaffected. Based on these observations, the ac-
tual procedure implemented for computing the regularization level in current version
of FDM2K is as following,
q2Act. = q2 M2
1 M22
∑
~n
|c(~n)|2 , (2.83)
where q2Act. is the actual value of regularization used in Eq. 2.82 and q2 is the relative
scale value specified by the user. Implemented this way, the optimal q2 for typical
experimental signals falls into a small range of from 10−5 to 10−3 with 10−4 being
optimal for most NMR experimental signals acquired under normal conditions: ≥
1mM sample concentration, normal probe, 500 MHz Spectrometer, 2 − 16 scans per
increment, room temperature, reasonably long T1 and T2, etc. Optimal q2 might be
lower for signals acquired under better conditions with higher SNR, and vice versa.
Several important aspects of FDM2K need to be discussed here. First, exactly
how particularly peaks are regularized by q is still not fully understood, due to the
highly nonlinear property of Eq. 2.82. In the original reference of FDM2K [74], we
demonstrated that in the simple case of a single peak, the peak position, intensity
113
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200
400
600
800
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200
400
600
800
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15N
/ Hz
15N
/ Hz
2D FT, N1xN2=300 x128 2D FDM2K, N1xN2=300 x8
b) q = 0.01
d) q = 0.16c) q = 0.04
a)
2D FDM2K, N1xN2=300 x8 2D FDM2K, N1xN2=300 x8
Figure 2.17: Chemical shift correlation spectra obtained by 2D FT and FDM2K (seetext). The signal used is the same as that used in Figure 2.15. The 2D FT spectrumwas obtained by using both N- and P-type data sets, each with N1 ×N2 = 300× 128complex data points. Cosine apodization functions were used in both dimensions.The signal length used in FDM2K calculations is N1 × N2 = 300 × 8. It shows thatwith appropriate level of regularization, stable and high-resolution spectra can beobtained using single FDM calculation (panel b). In addition, the behaviour of thespectra does have a smooth dependence on the regularization parameter (panel b, cand d). When the regularization is too large, the whole spectrum is further smoothedand small features are suppressed to baseline (panel d). However note that it alsohappens in a very non-uniform way. Also note that the values of q shown are relativevalues (see text).
114
and phase were not affected by the regularization while the regularized linewidth was
bigger than the true value. The change of linewidth is determined by the relative
amplitude of peak intensity dk and regularization level q: in the limit of q2 |d2k|,
the change is insignificant; in the other limit, the peak will be significantly broad-
ened. In the general cases of many resonances, how a particular peak is regularized
seems to depend not only on the peak intensity, but also on its environment such as
close-by peaks, degeneracy/near degeneracy with other peaks, and others. Second,
there is definitely some similarity between FDM2K and the conventional Tikhonov
Regularization. However, the regularization in FDM2K is efficient only when matri-
ces U1 and U2 are “bigger” than U014 . This is usually true as signals decay in the
time domain. However, in the case of noisy signals, some components of U1 or U2
may be larger than corresponding counterparts of U0 so that large q must be used in
order to suppress all artifacts, resulting in lower resolution. Figure 2.18 compares the
effects of Tikhonov regularization (implemented as RRT; see Section 2.5 for details)
and FDM2K using a 1D model signal. While the RRT spectra are regularized in
a smooth and predictable fashion, the FDM2K spectra show some unfavorable non-
uniform distortions. Third, matrices involved in Eq. 2.82 is asymmetric when q 6= 0.
To solve it exactly, both left and right eigenvectors should be computed. It is usually
accurate enough to calculate the right eigenvectors only, when q is small compared
14To understand this statement, one needs to write out the corresponding resolvent for Eq. 2.82for 2D spectral estimation, which is
R(ωl) ∼ (U†0U0 + q2) − eiτlωlU
†0Ul.
Therefore, q2 can guarantee that the resolvent is non-singular only when all eigen-components of U0
are greater than corresponding counterparts of Ul.
115
-180 -150 -120
1D RRT
-180 -150 -120
1D FDM2K
Lar
ger
Reg
ular
izat
ion
Frequency (Hz) Frequency (Hz)
Figure 2.18: Comparison of Tikhonov Regularization (implemented as RRT) andFDM2K using a “Jacob’s Ladder” model signal.
to the norm of the matrices and the eigenvectors are used only for calculating the
amplitudes. However, in other cases such as using the eigenvectors to construct some
nontrivial projection operators (e.g., J-projections; see Section 3.3), one might need
to compute both eigenvectors to avoid introducing extra artifacts.
2.4.4 Snap-Shot FDM: Self-Regularization.
There is another very interesting way of regularizing FDM where no additional
regularization parameters such as q is necessary. Instead, it requires experimentally
acquire multiple signals that consist of the same genuine peaks but uncorrelated noise.
In NMR experiment, multiple scans, c(i)(n), i = 1, 2, . . . , Ns, are routinely acquired
116
and then co-added to obtain an averaged signal, c(n) = 1Ns
∑Ns
i=1 c(i)(n), which has a
better signal to noise ratio. These scans can be conveniently stored separately before
they are co-added, even though this requires more disk space for data storage. The
second approach leads to little benefit if the Fourier transform is to be used. The
reason is that FT is a linear method. Whether FT is applied to each scan separately
and then the results are averaged or the signals are averaged before being transformed
does not change the resulting spectrum. However, when a nonlinear method like
FDM is used, storing each scan separately might be advantageous. It is possible to
co-process all scans together to distinguish “signal” and “noise” components, based
on the assumption that “signal” should be consistent from scan to scan while “noise”
is random and uncorrelated. The FDM algorithm can be modified to use the addition
information encoded in multiple scans, or “Snap-Shots”. This idea discussed below
was first suggested by Neumaier [95] and then exploited by Curtis [96]. We will use
1D as an example for the sake of clarity. 2D or higher dimensional cases are more
complicated but the basic idea is the same.
In the normal FDM, the problem of identifying the spectral parameters is re-
formulated as a generalized eigenvalue problem of Eq. 2.11 where the right and left
matrices are computed using the averaged signal. To process Ns signals that consist
of same set of genuine damped sinusoids, a super generalized eigenvalue problem can
be solved,
U(1)1
...
U(Ns)1
Bk = uk
U(1)0
...
U(Ns)0
Bk (2.84)
117
where U(i)l , l = 0, 1 are the evolution and overlap matrices constructed using ith
scan, c(i)(n). By solving all Ns GEPs in parallel for a single common set of eigen-
values and eigenvectors, components present in all data sets are emphasized while
the random (noise) components are suppressed. This technique has also been used
in radar imaging and acoustic signal processing [22, 3]. Similar techniques have also
been used to extract quantum tunneling splittings from semi-classical real-time cross-
correlation functions [97], or multiple excited vibrational states from imaginary-time
cross-correlation functions [78, 98].
The dimension of the rectangular matrices on both sides is NsKwin × Kwin, but
there will be only Kwin eigenvalues and corresponding eigenvectors. It is reasonable
and efficient to first reduce both matrices to Kwin × Kwin square matrices. There
are quite a few choices here. For example, one can simply apply SVD to right (or
left) matrix, then re-evaluate both matrices in the reduced space formed by Kwin
basis vectors that correspond to the Kwin largest singular values (see Section 2.4.2 for
details). Alternatively, one can use the FDM2K approach and recast Eq. 2.84 into a
normal GEP that only involves square matrices,
U †r Ul Bk = uk U †
r Ur Bk (2.85)
where Ur and Ul denote right and left rectangular matrices respectively. The second
procedure is numerically more efficient and generally yields similar results to the
truncated SVD approach.
By recasting the super GEP into a normal GEP, we can compute both eigenvalues
and eigenvectors using standard routines such as QZ or CG. The eigenvalues can be
118
more accurate than those computed either by using a single scan signal ci(n) or by
using the averaged signal. In particular, the complex frequencies that correspond to
“noise” tend to have a large and positive imaginary part, which means that the noise
spikes are effectively suppressed. However, it is less obvious how we can compute
the complex amplitudes as accurately. Remember that we first need to normalize the
eigenvectors with respect to BTk U0Bk′ = δkk′, where U0 is the overlap matrix, then
compute the amplitudes as√
dk = BTk C. It is not clear how we can use Ns copies
of overlap matrices U(i)0 and FT vectors C
(i)jointly in the most efficient way. There
are several formulas proposed and tested [96]. An expression that corresponds to the
average of all d(i)k was found to be optimal by numerical experiments,
√
dk =1
Ns
Ns∑
i=1
d(i)k =
1
Ns
Ns∑
i=1
Kwin∑
n=1[Bk]n C
(i)
n
√
B>k U
(i)0 Bk
(2.86)
There is another potential problem here. Compared to the averaged signal, the
SNR of each individual scan is√
Ns lower. Thus the matrices in the super GEP are
noisier, making the Snap-Shot FDM approach less attractive. Unlike the problem
of the ambiguity of computing the amplitudes, here we have a good solution, which
was also suggested by Neumaier [95]. Instead of using the noisy snap-shots c(i)(n)
directly, we first reconstruct a new set of snap-shots according to,
c(i)new(n) = c(n) +
1
Ns
(
c(i)(n) − c(n))
. (2.87)
The new signals have similar SNR to the average signal, while the random and un-
correlated noise is preserved. It has been shown by numerical experiments that this
procedure does improve the results [96].
119
Tested with simple 1D experimental NMR signals, Snap-Shot FDM seems to be
capable of separating “signal” from “noise”. The complex frequencies computed typ-
ically consist of certain number of genuine poles with relative small (true) line width
and a bunch of “noise” poles with much larger linewidth. The FDM ersatz spectrum
constructed tends to show a clean “signal” spectrum with a fat baseline (no noise).
Thus, the idea of “self-regularization” seems to work in these cases. However, when
the signal gets more complicated and requires local spectral analysis, the results were
less satisfactory. The self-regularization seems to be insufficient. When extended to
the most interesting and most demanding case of multi-dimensional spectral analysis,
the self-regularization seems to over-regularize the problem. Even genuine peaks are
significantly broadened. Amplitudes are also quite inaccurate due to the ambiguity
of the optimal formula. Nevertheless, the self-regularization is a very interesting idea,
deserving of further studies, which might have profound impact on deepening our
understanding of the optimal way of acquiring and processing time signals.
2.4.5 Summary
In this section, we discussed the ill-conditioning of multi-dimensional FDM and in-
vestigate various regularization techniques for suppressing the artifacts, among which
the FDM2K algorithm seems to be particularly efficient and effective, even though
by no means perfect or the best method for regularizing FDM. With regularization
implemented, it is possible to obtain stable, high-resolution spectral estimations in
a single FDM calculation, given that the signal sufficiently satisfies the Lorentzian
120
model and is long enough. The resulting spectrum has a smooth dependence on the
regularization level. The noise and small features are suppressed first. There is a
range of regularization level where the spectrum changes very little while the regular-
ization parameter q2 is changed by an order of magnitude. When the regularization
is too large, all features are significantly broadened but in a non-uniform way. How
a particular peak is affected by the regularization depends on its intensity as well as
it environments such as nearby peaks and (near) degeneracy with other peaks. The
problem of regularization of FDM is only partially solved at present. More studies
are needed for developing the ultimate method for regularizing FDM.
121
2.5 Regularized Resolvent Transform
There is a certain attraction to the idea of a transform: converting data from one
representation to another, often more useful, representation. For example, Fourier
Transformation converts a continuous function of time c(t) to a function of frequency
I(ω) by Fourier integration. For each frequency ω, we can obtain a complex number
that gives the local spectral amplitude I(ω). The simplicity and transparency of the
transform make it appealing even when, as in the discrete Fourier transform, the
digital spectrum may deviate appreciably from the true integral representation [7].
By contrast, parametric methods that rely on fitting the data to a functional form
are rather more complex in nature. There are typically adjustable parameters which
need to be tuned to obtain satisfactory results. In section, we bridge the parameter
estimation approach to the transform approach by introducing a new transform, the
Regularized Resolvent Transform (RRT): spectral representations are computed di-
rectly using the same data matrices as those in FDM without the intermediate step
of computing the spectral parameters. RRT maintains the spirit of local spectral
analysis and can be implemented efficiently.
2.5.1 One-Dimensional RRT
In this section we consider a 1D spectral analysis problem which has been pre-
viously treated by a variety of methods. However, the expressions derived here are
generalized to the much less explored multidimensional cases in the next section.
Given a discrete equidistant time signal c(n) ≡ c(nτ), the goal is to estimate its
122
infinite time DFT, defined in Eq. 2.26, but using only a finite portion of the data
with n = 0, 1, ..., N − 1. The assumption used here to derive the linear algebraic
expressions is the same as the quantum ansatz used in FDM: c(n) is associated with
a time autocorrelation function of a fictitious dissipative quantum system defined by
an effective evolution operator U and some initial state Φ0 [24],
c(n) =(
Φ0|UnΦ0
)
. (2.88)
The assumption that the evolution operator U has a finite rank is equivalent to as-
suming that the signal of question can be represented as a sum of damped sinusoids.
Substituting Eq. 2.88 into Eq. 2.26 and evaluating the geometric summation analyt-
ically, we obtain:
I(ω) = τ
(
Φ0
∣
∣
∣
∞∑
n=0
einτωUn − 1
2
Φ0
)
= τ
(
Φ0
∣
∣
∣
1
1 − eiτωU− 1
2
Φ0
)
. (2.89)
Note that U∞ vanishes as the system is assumed to be dissipative.
Evaluated in an appropriate basis |Ψj), the operator expression of Eq. 2.89
becomes a working formula for directly calculating I(ω),
I(ω) = τ
[
CTR(ω)−1C − c(0)
2
]
, (2.90)
with resolvent matrix R(ω) = U0−eiτωU1. The elements of the evolution and overlap
matrices are defined respectively as, [U1]jj′ = (Ψj|UΨj′) , [U0]jj′ = (Ψj |Ψj′) , and the
coefficients of the column vector C are [C]j = (Ψj|Φ0). In particular, when a Fourier-
type basis defined in Eq. 2.15 is used, the spectral properties around some frequency
ω are completely defined by a very small subspace |Ψj) of size Kwin with ϕj ∼ ω.
123
Therefore, only a small Kwin × Kwin matrix R(ω) has to be inverted in Eq. 2.90 to
yield an accurate spectral estimation, I(ω). The expressions for computing matrix
elements are already given in Section 2.2 (Eq. 2.36 and Eq. 2.37). The column vector
C is FT of the original signal array c(n),
[C]j =M−1∑
n=0
einτϕjc(n) . (2.91)
Clearly, evaluation of Eq. 2.90 can be done for all values of ω within a chosen
frequency window by solving the corresponding generalized eigenvalue problem to
obtain the eigenvalues uk and eigenvectors Bk, which is the approach used in FDM.
Although, this method might seem preferable to any other alternative, we note that
Eq. 2.90 can also be evaluated directly, for example, by solving the associated linear
system,
R(ω)X(ω) = C , (2.92)
and then using
I(ω) ≈ τ
[
CTX(ω) − c(0)
2
]
. (2.93)
Mathematically, both approaches should provide exactly the same results. However,
numerically, the second approach might have advantages due to its simplicity and
transparency. In addition, Eqs. 2.92 and 2.93, may actually appear computationally
efficient if the spectrum is to be evaluated at relatively few values of ω. This will be
even more the case for the multi-dimensional spectral estimation.
A less obvious issue is how stable and robust the algorithm is. Apparently, the
matrix R(ω) may be very ill-conditioned, so its inversion or use in Eq. 2.92 requires
some kind of regularization. One possibility is to use SVD of R(ω) to calculate a
124
pseudo inverse (see Section 2.4.2: soft regularization). However, SVD, if applied for
each value of ω, would be quite expensive. A much less expensive regularization of
the resolvent can be obtained using the Tikhonov Regularization [93, 99],
I(ω) ≈ τ
[
CT (
R(ω)†R(ω) + q2)−1
R(ω)†C − c(0)
2
]
, (2.94)
where the dagger † means Hermitian conjugate and q is a real regularization param-
eter. With such a regularization the singularity in the denominator is removed as(
R†R + q2
)
is a Hermitian and positive definite matrix.
Eq. 2.94 can be evaluated by solving the regularized Hermitian least squares prob-
lem,(
R†(ω)R(ω) + q2
)
X(ω) = R†C , (2.95)
and then using Eq. 2.93.
It should be noted that the expensive matrix-matrix multiplication R(ω)†R(ω) in
Eq. 2.95, which is a K3win process, does not have to be performed at each value of ω.
Significant numerical saving could be achieved by using
R(ω)†R(ω) = U†
0U0 + U†
1U1 − e−iτωU†
1U0 − eiτωU†
0U1 .
The spectral estimation given by Eq. 2.94 is one of the main results of this section.
Operationally it has a status of “transform” (like DFT), while a “method”, e.g. the
Filter Diagonalization Method, would refer to a procedure that would generally be
less obvious to use. More precisely, Eq. 2.94 corresponds to a direct nonlinear trans-
formation, here called the Regularized Resolvent Transform (RRT), of the time signal
to the frequency domain spectrum. Unlike most other nonlinear high resolution spec-
tral estimators, RRT is very stable, computationally inexpensive, and has adjusting
125
parameters that are straightforward to use. These parameters are the size, Kwin,
and density, ρ, of the Fourier basis in the frequency domain, and the regularization
parameter q. Note that Kwin could in principle be as small as 3, although a larger
Kwin generally improves the resolution, while increasing the CPU-time according to
the cubic scaling of a linear solver. For sufficiently large Kwin, which is usually less
than 100, the results do not change noticeably. Thus, Kwin can be chosen according
to how long one would like to wait for the spectrum to be computed. The choice for
the basis density parameter, ρ, between 1.1 and 1.2 usually works well if a single-scale
basis (as opposed to the multi-scale one [79]) is used.
The spectrum I(ω) computed by RRT with q = 0 should generally be indistin-
guishable from that computed by the FDM algorithm based on solving the generalized
eigenvalue problem, Eq. 2.11, and using Eq. 2.27 for spectral estimation. However, for
q > 0 they will differ as demonstrated in Figure 2.19 using the model signal, “Jacob’s
Ladder”, described in Section 2.2. The signal contains 50 triplets of which both the
peak widths and the splittings gradually decrease from the right to the left, allowing
examination the breakdown of the resolution for any fixed signal size. The spectral
region shown has very small spacings and is therefore hard to resolve. The upper trace
is the exact spectrum that could be obtained by RRT using, for example, N = 48K
data points. The DFT spectrum using N = 64K cannot resolve all the triplets in this
region, while using DFT with N = 32K does not resolve any of them. In addition,
one can see some baseline ringing that is not completely removed by apodization of
the signal. The RRT result using N = 32K and q = 0 is very close to the exact result,
although some linewidths and amplitudes are slightly inaccurate. Interestingly, one
126
3.5 4.0 4.5
N=64KExact
N=32K
RRTq=0
RRTq=3x10
-5
RRTq=3x10
-4
DFT
N=24K
RRTq=0
RRTq=6x10
-6
HzFigure 2.19: Absorption spectra obtained by processing a model time signal, “Jacob’sladder”, by the Regularized Resolvent Transform (RRT) and DFT using different sizesof the signal, N = 32, 000 and N = 24, 000. The RRT spectra have much higherresolution than the DFT spectrum with N = 32, 000. The non-regularized spectra(q = 0) may be quite non-uniform. An increase of the regularization parameter qgenerally leads to a more uniform spectral estimate and gradual decrease of resolution,however the best result is often obtained using RRT with some small q > 0. The qvalues shown are relative values. The actual value used in Eq. 2.94 is computed asq2act = q2 < [R†R]ii >i.
127
can control the appearance of the RRT spectrum by tuning q. An increase of q makes
the spectral estimation more uniform while decreasing the resolution. Note that the
scale for q is related to the normalization of the U-matrices.
The lower two traces show RRT spectra using N = 24K, which is too short for
the method to resolve the narrowest multiplets in this spectral region. The resolution
failure for q = 0 manifests itself in a quite non-uniform appearance of the peaks in
the multiplets. In particular, only two peaks appear in the left most triplet with the
smallest spacings. By increasing q one can make them more uniform and sometimes
improve the resolution (e.g., note the triplet at ∼ 22 Hz). That is, for short data
sets the RRT “fails” in a controllable fashion (as is the case for DFT), while typically
providing a higher resolution than DFT for narrow Lorentzian lines.
It is important to note here a subtlety present in all linear algebraic algorithms, in
particular, in FDM and RRT, related to the fit of the time signal by damped sinusoids,
Eq. 2.1. For example, in RRT the derivation of Eq. 2.89 assumes convergence of the
infinite geometric series∑∞
n=0
(
eiτωU)n
. However, this assumption may be ambiguous
since only a finite part of the data is available. Numerically, when evaluated in a finite
basis U may have eigenvalues uk = e−iτωk outside the unit circle, corresponding to
the negative linewidth γk. Section 2.2.3 contains a detailed discussion of how to
deal with these “spurious” poles. However, in RRT, the spectral parameters are not
explicitly computed, making direct manipulation of individual poles impossible. This
has both advantages and disadvantages. One one hand, we can conserve the mutual
interferences and cancellations to the maximum extend; on the other hand, when the
true peak width is very small or zero as in Constant-Time NMR experiment (see
128
γ=0.1Hz γ=0.01Hz γ=−0.01Hz γ=−0.1Hz
~ ~absorption
dispersion
Figure 2.20: The real (absorption) and imaginary (dispersion) parts of the RRTspectrum I(ω) = τdk/(1 − eiτ(ω−ωk)) for a single line as a function of its “width”γ = −Im ωk. The sign of the absorption peak is flipped when γ changes its sign,while the appearance of the dispersion peak only depends on the absolute value of γ.
Chapter 3 for more details), it is likely to have γk < 0 due to noise and/or numerical
errors. This in turn results in a flip of the sign of the absorption part of the peak as
shown in Figure 2.20. Note though that the sign of the dispersion part is unaffected.
This property of the RRT lineshapes can be easily understood by considering the
behavior of I(ω) (see Eq. 2.27) near an eigenfrequency ωk = ωr − iγ, i.e., assuming
τ |ω − ωk| π and then extracting the real and imaginary parts of the complex
Lorentzian as
I(ω) ≈ τdk
1 − eiτ(ω−ωk)≈ idk
ω − ωk=
dkγ
(ω − ωr)2 + γ2
+ idk(ω − ωr)
(ω − ωr)2 + γ2
. (2.96)
Since in RRT the eigenvalues are not computed, one cannot manipulate with those
in a simple fashion like in FDM. One way to circumvent this problem in RRT is to
shift the argument of I(ω) by iΓ with Γ > −γk , i.e, construct I(ω + iΓ). It replaces
γk of all peaks by γk + Γ and effectively flips the negative γk. The result of such
a shift is demonstrated in the third trace of Fig. 2.21 where all the peaks have the
129
correct sign, but are slightly broadened. Clearly, this procedure may not be always
well defined and is one drawback of RRT. For example, peaks with γk < −Γ are
effectively narrowed and might result in artifacts.
360 380 400 Hz
Exact
N = 300
RRT, Γ= 0 Hz
RRT, Γ= 1 Hz
DFT
absorptiondispersion
Figure 2.21: An example of a failure of the RRT spectral representation with no shiftand no regularization implemented (second trace, both q = 0 and Γ = 0). Because ofa too short signal used to construct the spectrum both the amplitudes and widths ofthe peaks are quite inaccurate, in particular, one peak appears with the wrong sign(note that the appearance of the dispersion peaks is correct). The wrong absorptionpeak is flipped by plotting I(ω+ iΓ) (the lowest trace) instead of I(ω) with Γ = 1 Hz.Note, that this procedure replaces γk of all peaks by γk + Γ, i.e., effectively fixes thesigns of all peaks with γk > −Γ and broadens all the peaks with the correct γk > 0.
130
2.5.2 Multi-Dimensional RRT
Generalization of RRT to a multidimensional case is essentially equivalent to how
the 1D FDM is generalized. Since there is only a minor difference between the 2D
RRT and D > 2 RRT, for the sake of simplicity, we restrict our presentation to the
D = 2 case.
To this end, given a 2D time signal c(~n) ≡ c(n1τ1, n2τ2) defined on an equidistant
rectangular time grid, our goal is to estimate the 2D (infinite time) discrete Fourier
sum, defined in Eq. 2.58, using only the finite N1 × N2 part of the signal. The 1D
quantum ansatz of Eq. 2.88 is generalized by using two commuting complex symmetric
evolution operators U1 and U2,
c(~n) ≡ c(n1τ, n2τ2) =(
Φ0|Un11 Un2
2 Φ0
)
. (2.97)
Inserting Eq. 2.97 into Eq. 2.58 and evaluating the geometric sums analytically gives
I(ω1, ω2) = τ1τ2
(
Φ0
∣
∣
∣
∣
∣
1
1 − eiτ1ω1U1
− 1
2
×
1
1 − eiτ2ω2U2
− 1
2
Φ0
)
. (2.98)
Evaluated in the 2D Fourier basis, defined in Eq. 2.50, the operator equation of
Eq. 2.98 becomes the working resolvent transform expression for estimating complex
infinite time 2D DFT spectrum,
I(ω1, ω2) ≈ τ1τ2 (2.99)
×
CT[
R1(ω1)−1U0 R2(ω2)
−1 − R1(ω1)−1 − R2(ω2)
−1
2
]
C +c(0, 0)
4
,
with resolvent matrices
Rl(ωl) = U0 − eiτlωlUl , l = 1, 2 , (2.100)
131
where Ul matrices and the Fourier vector C can be evaluated using the expressions
derived previously in Section 2.3 (Eqs. 2.52, 2.53, 2.54, and 2.57).
A 2D RRT can be obtained by regularizing the two resolvents in Eq. 2.99, leading
to
I(ω1, ω2) ≈ τ1τ2
X(ω1)TU0X(ω2) −
X(ω1)TC + C
TX(ω2)
2+
c(0, 0)
4
, (2.101)
with the two frequency-dependent vectors Xl(ωl), l = 1, 2, computed by solving the
regularized Hermitian least squares problems,
(
Rl(ωl)†Rl(ωl) + q2
)
Xl(ωl) = Rl(ωl)†C . (2.102)
Note that the total number of the linear systems to be solved for each 2D frequency
window is equal to Nω1 + Nω2 instead of Nω1 × Nω2 , where Nω1 and Nω2 are the
numbers of the frequency grid points, ω1 and ω2, used to plot the 2D spectrum in the
window.
Derivation of the 2D RRT is another main result of this section. Similar to the
2D FDM, its substantial advantage as compared to the 2D DFT is in the ability
to process the whole 2D data set simultaneously, thus converting the time domain
information into the frequency domain with minimal loss. Unlike the 2D FDM, in
RRT we do not directly refer to the 2D harmonic inversion problem, although the
two methods are very closely related. In particular, if the form of Eq. 2.42 is satisfied,
the RRT spectrum for a sufficiently large data set processed should converge to
I(ω1, ω2) = τ1τ2
K∑
k=1
dk
2
2
[1 − eiτ1(ω1−ω1k)] [1 − eiτ2(ω2−ω2k)](2.103)
− 1
[1 − eiτ1(ω1−ω1k)]− 1
[1 − eiτ1(ω2−ω2k)]
+τ1τ2 c(0, 0)
4.
132
Ideally, the results should fully converge if the total size of the data set satisfies the
condition Ntotal = N1 × N2 ≥ 4K, no matter whether N2 or N1 is small. This, of
course, does not take into account noise, degeneracies and roundoff errors which could
affect the convergence conditions significantly.
Due to its numerical stability, robustness and ease implementation, the RRT has
an advantage over FDM, when only complex 1D or 2D FT type spectra, as defined
by Eqs. 2.26 and 2.58, are of interest. Note, yet, an advantage of FDM, especially, in
the multidimensional case, lies in its ability to construct various nonanalytic spectral
representations, such as a double-absorption spectrum using a single purely phase
modulated 2D data due to the availability of the spectral parameters (Eq. 2.68). Fur-
thermore, in FDM inaccuracies in determining the imaginary parts of the computed
frequencies, in particular, their signs, might be corrected. In RRT the spectral pa-
rameters are not produced and therefore the artifacts are removed by regularization
with a finite value of q and/or by shifting the spectrum using I(ω1 + iΓ1, ω2 + iΓ2)
with shifting parameters Γ1 and Γ2.
In the next example we apply 2D RRT to the process the same 15N-1H HSQC
signal of rubredoexin used in previous examples. Figure 2.22 compares the RRT and
DFT spectra obtained using various signal sizes. Even though this signal is particular
favorable for DFT as it contains pure singlet of similar intensity, RRT still provides
a welcome improvement in resolution or a decrease in experimental time for given
resolution. As in FDM, the gain in resolution hinges on the total information content
of the signal, which depends on signal size, signal to noise ratio, amount of various
imperfections and others. It is neither possible to improve the resolution of very noisy
133
spectra, nor to magically divine the presence of peaks buried in noise.
FFT N2=128 N1=512
-600-400-2000200400600
-600
-400
-200
0
200
400
600
FFT N2=16 N1=512
-600-400-2000200400600
-600
-400
-200
0
200
400
600
RRT N2=6 N1=300
-600-400-2000200400600
-600
-400
-200
0
200
400
600
RRT N2=16 N1=300
-600-400-2000200400600
-600
-400
-200
0
200
400
600
Figure 2.22: Chemical shift correlation spectra of the metalloprotein rubredoxin ob-tained by DFT and RRT using various signal sizes. All the double-absorption spectraare generated using the conventional procedure which combines the complex N- andP-type complex spectra. Regularization parameters are optimized for both RRT cal-culations. The shifting parameters Γ1 = Γ2 = 1 Hz.
134
2.5.3 Other Spectral Representations Using RRT
The resolvent formulas of Eq. 2.90 and Eq. 2.99 provide us a way to estimate
the infinite time DFT spectra using only finite data sets. In this section, I would
like to demonstrate that RRT is a more powerful method than just nonlinear FT-
type spectral estimator. With minor modifications, the resolvent formulas can be
used to construct many other types of useful spectral representations, revealing more
information about the signal.
1. Inverse Laplace transform via RRT: iRRT
Any complex variable can be used in RRT. In particular, when the frequency
variable is purely imaginary, RRT becomes an efficient expression for computing in-
verse Laplace transform without exponential instabilities. It has been shown that
it is possible to recover up to 4 exponential decaying constants with significant gap
between the fastest and lowest decaying components in the Quantum Monte Carlo
(QMC) calculations of vibrational states of polyatomic systems [98]. The RRT ex-
pression with imaginary frequency variable is also called iRRT. iRRT has been also
applied to processing the Diffusion Ordered Spectroscopy (DOSY) [100, 101] NMR
experimental signals, showing the promise to identifying multiple diffusion constants
even in the heavily overlapped spectral regions [96]. However, the overall results have
not been totally satisfactory considering the reliability of the results and accuracy
of intensities. There are several practical difficulties: first, accurate characterization
of both fast and slow decaying components requires signals with very high signal to
noise ratio, while typical DOSY experimental data sets are very noisy due to the lim-
135
ited quality of magnetic field gradient and many other experimental limitations [102].
Second, fitting data to multiple exponential decaying components has been proved
to be a notoriously unreliable procedure. Even though the iRRT expression seems
to be very stable, there are still uncertainties such as the effect of regularization on
the accuracy of computing the decaying constants. It is necessary to study the re-
sults as a function of the signal length (or other parameters such as basis size and
regularization) to check the validity of the results. Finally, the intensity computed
by iRRT is not accurate enough for quantitative usage and it is not obvious how it
can be improved.
2. Pseudo-2D RRT spectra of both real and imaginary frequencies
There is another an interesting spectral representation that can be constructed
using RRT with complex variables. By computing the spectral intensity I(ω + iΓ)
as a function of both the real frequency ω and imaginary frequency Γ, a pseudo-2D
spectrum results. As I(ω + iΓ) diverges at ω + iΓ = ωk on the complex plane, we can
identify both the peak positions and the linewidths simply by looking at the singular
points on the 2D contour plot. In addition, it is also possible to separate peaks that
are severely overlapped according to their linewidths. Figure 2.23 shows a contour
plot of such a pseudo-2D spectrum, applied to a 1D NMR signal. Note that a slice
along Γ = 0 Hz corresponds to the normal 1D FT-type absorption spectrum with
no smoothing, which is shown on top of the 2D plot for direct comparison. The 2D
plot reals more information about the signal. For example, the 2D plot shows that
there are possibly two peaks, A and B, at around ω = −5600 Hz (indicated by two
arrows), while this information is totally hidden in the conventional 1D FT spectrum
136
due to the heavy overlapping.
-6400 -6200 -6000 -5800 -560015
10
5
0
-5
-10
Frequency (ω) / Hz
Lin
e W
idth
(−γ) / H
z
1D FT
Re[Ι(ω+iγ)] Α
Β
ΑΒ
Figure 2.23: Contour plot of I(ω + iΓ) on the complex plane. Peaks are spreadout along the imaginary frequency axis according to the line width, providing moreinformation than the conventional 1D FT spectrum (shown on the top). For example,the pseudo-2D plot indicates that there are possibly two peaks, noted as A and B,around -5600 Hz, while from the 1D FT spectrum this information is hidden due tothe heavy overlapping of the two peaks. Note that a slice at γ = 0 of the pseudo-2Dspectrum corresponds to the 1D FT-type spectrum without any linebroadening.
2. Pseudo-absorption 2D RRT spectral estimations
When severely truncated data sets are used in RRT, just like in FDM, while the
positions of the peaks converge best, the phases of the complex amplitudes are least
137
accurate, which manifest themselves as the incorrect phases of corresponding peaks
in the double-absorption spectrum obtained by combining N- and P-type complex
spectra. For example, close examination of the RRT spectra shown in Figure 2.22
reveals several peaks slightly out of phase, e.g., the peak at around (-600 Hz, 350 Hz).
In principle, increasing regularization will suppresses the phase instability. However,
numerical experiments showed that the phase stability improved slowly with regard to
RRT regularization, especially for truncated signals. Moreover, in certain situations,
only a single purely phase modulated data set is available (e.g., in 2D J experiments).
In such cases an absolute value RRT spectrum may be an appealing and/or the only
option, although the resolution of the latter are unacceptable due to the contribution
of the dispersion lineshapes. Note, yet, there exists a variety of other spectral rep-
resentations, in which the dispersion contributions are eliminated, therefore leading
to much higher resolution. An example of such a representation in 2D, which is also
used in 2D FT spectral analysis, is
A(ω1, ω2) = ReI(ω1, ω2 + iΓ2) − I(ω1, ω2 + iΓ′2) . (2.104)
The dispersion contributions along ω2 are eliminated by the subtraction of the two
complex spectra using different shifts Γ2 6= Γ′2 (with possibly positive Γ2 and negative
Γ′2), while taking the real part leads to the absorption lineshapes along ω1. The
disadvantage of Eq. 2.104 is the need to fiddle with too many adjusting parameters.
Another very useful spectral representation is given by
I(2)(ω1, ω2) =
Φ0
∣
∣
∣
∣
∣
∣
∣
τ1τ2[
1 − eiτ1ω1U1
]2 [
1 − eiτ2ω2U2
]2Φ0
138
-600-400-2000200400600
-600
-400
-200
0
200
400
600
Figure 2.24: RRT pseudo-absorption spectrum∣
∣
∣I(2)(ω1 + iΓ1, ω2 + iΓ2)∣
∣
∣ (see
Eq. 2.105) constructed using only the N-type (i.e., the purely phase modulated) dataof Figure 2.22 with N1 = 16, N2 = 300, q = 0.0003, Γ1 = 2 Hz and Γ2 = 1 Hz.
=K∑
k=1
τ1τ2dk
[1 − eiτ1(ω1−ω1k)]2[1 − eiτ2(ω2−ω2k)]
2 . (2.105)
The absolute value spectrum∣
∣
∣I(2)(ω1, ω2)∣
∣
∣ will have a great resolution advantage
over |I(ω1, ω2)|. The peaks are narrow and have quasi-absorption lineshapes in
∣
∣
∣I(2)(ω1, ω2)∣
∣
∣. Note that the amplitude dk is not squared. Thus the peak integral
will effectively be divided by the product of the linewidths in two dimensions, γ1kγ2k.
Accordingly, I(2)(ω1, ω2) will favor the narrow peaks against the broad ones. This non-
uniform distortion can be reduced by shifting the spectrum using∣
∣
∣I(2)(ω1 + iΓ1, ω2 + iΓ2)∣
∣
∣
which effectively increases the widths of all peaks making them more uniform. While
I(2)(ω1, ω2) cannot be constructed by FT, it is representable in terms of the U-
matrices in complete analogy with Eq. 2.99. Of course, the corresponding resolvents
139
should be regularized accordingly. In Figure 2.24 we show the RRT spectra using
∣
∣
∣I(2)(ω1 + iΓ1, ω2 + iΓ2)∣
∣
∣ with Γ1 = 1 Hz and Γ2 = 10 Hz using the same 2D signal
as in Figure 2.22. The peaks do have absorption lineshapes, although the heights
are somewhat distorted when compared to the correct spectra of Figure 2.22. Higher
order pseudo-absorption spectra can also be easily constructed by using higher power
of the resolvent matrices. In the later cases, SVD is preferred, as once decomposi-
tion of the resolvent matrix R(ωl) is available, it is trivial to construct regularized
pseudo-inverse matrix R−nq (ωl) with same computational efforts for any integer n.
2.5.4 Extended Fourier Transform
As we discussed before, the phase computed by RRT seems to be the least stable.
Especially when the signal is severely truncated in all dimensions (even though it
rarely so in NMR experiments), the RRT spectra could be significantly distorted.
Increasing the regularization does not help much here unless the regularization level
is so high that most of the features are smoothed out. This is a disadvantage of RRT.
On the other hand, even though FT converges slowly with regard to the signal size, it
does not fail even in the severely truncated cases. The eXtended Fourier Transform
(XFT), developed by Armstrong and Mandelshtam [103], is a hybrid method of DFT
and RRT designed to combine the high resolving power of RRT and the stability of
DFT. The basic idea is to use RRT only to estimate the remainder of finite DFT of
the signal. The details of XFT can be found in the original reference [103]. Here we
give a brief derivation of 1D XFT for the illustration purpose.
140
Given a finite time signal c(n) defined on an equidistant time grid of N points,
we want to estimate the infinite time DFT spectrum by breaking it into two parts15,
I(ω) = τ∞∑
n=0
c(n)eiωnτ = τ
N−1∑
n=0
c(n)eiωnτ +∞∑
n=N
c(n)eiωnτ
. (2.106)
Note that the first summation is simply the conventional finite DFT summation. Due
to the truncation of the signal, the second term can not be directly computed, which
is the source of FT time-resolution uncertainty principle and the sinc-like truncation
artifacts (see Chapter 1). However, it can be estimated using the RRT expression,
∆N (ω) =∞∑
n=N
c(n)eiωnτ =∞∑
n=N
eiωnτ(
Φ0
∣
∣
∣Un∣
∣
∣Φ0
)
=∞∑
n=N
eiωnτ(
ΦN/2
∣
∣
∣Un−N∣
∣
∣ΦN/2
)
= eiωNτ∞∑
n′=0
(
ΦN/2
∣
∣
∣Un′
eiωn′τ∣
∣
∣ΦN/2
)
= eiωNτ
(
ΦN/2
∣
∣
∣
1
1 − eiτωU
∣
∣
∣ΦN/2
)
. (2.107)
Evaluated in the Fourier basis defined in Eq. 2.15, we can obtain the numerical
expression with regularization for estimating the remainder of DFT,
∆N (ω) = eiωNτ CT
N/2 R−1
q (ω) CN/2 , (2.108)
with the same resolvent matrix R(ω) = U0 − eiτωU1 and a shifted FT vector,
[CN/2]j =M−1∑
n=0
einτϕj
(
Φn|ΦN/2
)
=M−1∑
n=0
einτϕjc(n + N/2) . (2.109)
Therefore, XFT is essentially a straightforward extension of RRT. We simply need
to construct the shifted FT vector C, evaluate the RRT estimation of the DFT re-
mainder, then add the correction to the DFT summation. Figure 2.25 is an interesting
15The zero correction term is dropped out here for the sake of simplicity.
141
-700 -650 -600 Hz
XFT
DFT (no apodization)
Correction
DFT N=32000
DFT (apodized)
N=5000
Figure 2.25: An interesting region of the 1D NMR spectrum of 12H-pyrido[1,2-a:3,4-b]diinole, processed by the XFT using a truncated (N=5000) data set. The figureillustrates the behavior of each term involved in the calculation of the XFT spectrum.The XFT spectrum is obtained by adding together the unapodized DFT term, andthe correction term calculated with a regularization parameter of q that is optimizedfor this case. The correction term exactly cancels the sinc-like oscillations presentin the unapodized DFT. It also adds small terms to increase the resolution of somepeaks, allowing some doublet structures to be identified. For comparison the apodizedDFT spectra for N=32000 and N=5000 are presented. (This figure is courtesy of G.S. Armstrong [103]).
142
illustration of how XFT works. The XFT of an experimental NMR signal is used to
examine the roles of both terms in Eq. 2.106. The lower three traces were computed
with only the first 5000 data points. The DFT trace shows several unresolved multi-
plets and sinc-oscillations in the baseline. The correction trace has oscillations equal
and opposite to those in the DFT trace. When added together, the oscillations cancel,
giving the correct baseline in the XFT trace. Furthermore, the correction contains
some high resolution features, revealing some splittings that are not resolved in the
DFT spectrum.
It is less obvious but can be proved that when the signal satisfies the Lorentzian
model and is sufficiently large, the XFT spectrum, Eq. 2.106, gives the exact infinite
time DFT of the signal. By separating the infinite DFT summation into two parts
and evaluating them independently provides us a way to combine the stability of DFT
and high-resolution of RRT together: on one hand, one can use a large regulariza-
tion to suppress the contribution of RRT correction, providing a more conservative,
finite DFT like spectrum; on the other hand, one can optimize the regularization for
maximum resolution enhancement. Thus, XFT does not fail even when the signal is
too short to obtain meaningful high-resolution spectrum. This is a very attractive
property, especially considering that in FDM and RRT, it is either “everything” or
“nothing” for certain cases.
XFT can be also extended to processing multi-dimensional data sets. The idea is
also to use RRT only for estimating the remainder of 2D DFT. Details can be found
in Ref. [103]. Applied to experimental 2D NMR signals, it has been shown that even
for signals severely truncated in both dimensions, 2D XFT can provide very stable
143
2D spectral estimation with resolution similar to conventional 2D DFT but without
any sinc-like truncation artifacts. In the case of large data sets, 2D XFT is able
to provide higher resolution beyond FT uncertainty principle. However, the phase
instability seen in 2D RRT is still present in high-resolution the 2D XFT spectra,
which is a little disappointing but reasonable as RRT is responsible for the high-
resolution features. In addition, the shift FT vector CN/2 is computed using the tail
of the signal and therefore more noisy than the FT vector used in RRT, making the
phase instability potentially more severe in the XFT high-resolution estimation.
2.5.5 Conclusions
In this section, we derived a new numerical expression, the Regularized Resolvent
Transform (RRT), which corresponded to a direct transformation of the time-domain
data to a frequency-domain spectrum. RRT is suitable for high resolution spectral
estimation of multidimensional time signals. One of its forms, under the condition
that the signal consists only of a finite number of damped sinusoids, turns out to be
equivalent to the exact infinite time discrete Fourier transformation. RRT naturally
emerges from the Filter Diagonalization Method (FDM), but no diagonalization is
required. The spectral intensity at each frequency ω is expressed in terms of the
resolvent R(ω)−1 of a small data matrix R(ω) constructed from the time signal.
Generally R is singular, and thus its inversion requires certain regularization. Among
various possible regularizations of R−1, the Tikhonov regularization appears to be
computationally both efficient and stable. Numerical implementation of RRT is very
144
inexpensive because even for extremely large data sets the matrices involved are small.
We also showed that RRT could be used to constructed various non-FT spectral
representations. In particular, using a purely imaginary frequency variable, RRT
becomes an expression for estimating inverse Laplace transform without exponential
instabilities. RRT can also be used to compute a pseudo-2D spectrum, I(ω+iΓ), with
respect to both the real and the imaginary frequencies, revealing more information
than the conventional FT-type spectral estimation. Finally, pseudo-absorption type
spectra can be also constructed by RRT when an absolute value spectrum is desirable.
When the signal is severely truncated in all dimensions, 2D RRT spectra might
be highly distorted due to the phase instability. Even though it can be improved by
constructing a pseudo-absorption spectrum, there exists a more elegant alternative.
The eXtended Fourier Transform, or XFT, attempts to combine the stability of DFT
and high-resolution of RRT, by using RRT only to estimate the remainder of finite
time DFT. XFT can provide a stable high-resolution estimation when the data set is
sufficiently large and does not fail even the data set is extremely small.
145
2.6 Reliability and Sensitivity of FDM/RRT
In previous sections, we have been focusing on how to extract maximal informa-
tion from the available data. In this section, we will exploit another important aspect
of signal processing—-checking the reliability of the results, which has been ignored in
previous discussions. Unfortunately, this is no easy task due to the high nonlinearity
and complexity of FDM and RRT algorithms. What makes the analysis even more
complicated are various practical factors including noise, lineshape distortions and
other NMR experimental artifacts16, which are very difficult to model and quantify.
As a result, we will only be able to check the reliability of FDM/RRT results in var-
ious qualitative and/or semi-quantitative ways. Instead of providing some rigorous,
conclusive statements, we will focus on how to double check the results and make
some qualitative judgments. A semi-quantitative way for studying the sensitivity of
FDM/RRT spectral estimations with respect to various processing parameters will
also be described.
2.6.1 Reliability of 1D FDM and RRT
Reliability of 1D FDM and RRT is relatively easy to check. As we known, both
FDM and RRT can be used as a high-resolution spectral estimator. One of the
simplest ways to check the validity of the result is to inverse DFT the FDM/RRT
spectrum to recover the time domain signal and compare it with original one. Al-
ternatively, when FDM is used as a parametric method, one can plug the parameter
16Such as t1 noise, pulsing artifacts and phase-cycling/gradient residual artifacts
146
-1000 -500 0 500 1000
0 0.5 1 1.5 2
DFT
FDM with Nb=50, Nbc=10
Difference
Frequency (Hz)
Time Signal c(t)
Re[c(t)]
Im[c(t)]
Time (second)
Figure 2.26: The spectral estimation obtained by FDM matches almost perfectly withthe DFT spectrum. The difference is within the noise level except for one of the threemost intensive peaks. The parameters used in FDM calculation were set to typicalvalues and not optimized.
list ωk, dk into Eq. 2.1 to reconstruct the time signal and compare it with the orig-
inal one. In the examples given in Section 2.2 and 2.5.1, we already compared the
FDM/RRT spectra with DFT spectra and showed that they matched very well. Here
we present a reliability study where FDM is used both as a parametric method and as
a spectral estimator to analyze a noisy experimental NMR signal. Figure 2.26 com-
pares the FT and FDM spectra, with the inner panel showing the real and imaginary
part of the time domain signal. Even though the signal is very noisy and contains
many heavily overlapped features, the difference between DFT and FDM spectra is
147
0 0.5 1 1.5 2
Real Part
0 0.5 1 1.5 2
Imaginary Part
Time (second)Time (second)
Original Signal
Reconstructed Signal
Difference
Original Signal
Reconstructed Signal
Difference
Figure 2.27: Comparison of the original signal and that reconstructed from the spec-tral parameters computed by FDM. Due to the use of multi-scale basis and multi-windowing, only an approximate complete line list that primarily describes narrowfeatures can be obtained (see text), and used here to reconstruct the time signal.The initial part of the signal is not well reproduced. However, the rest of this signal,from t = 0.2s up to t = 2s (maximum), mainly defined by narrow poles, is very wellreproduced. The difference between original and reconstructed signals is within thenoise level.
within the noise level throughout the spectral range except a region around one of
the three most intensive peaks (∼-50 Hz). Similar comparison can be made between
RRT and DFT spectra (data not shown). FDM/RRT have also been tested using
many other 1D NMR experimental signals and proved to be very stable and reliable
as high-resolution spectral estimators.
It is less straightforward to check the spectral parameters computed by FDM
148
for experimental signals. One of the reasons is that the “true” spectral parameters
are not available for direct comparison. In addition, the spectral parameters are
the most accurate when a multi-scale Fourier basis is used, where the background is
recomputed for each window calculations. It is not obvious how to combine results
from all window calculations to construct an effective line list for describing the
background (very broad poles). Furthermore, the adjacent spectral windows overlap
with each other by 50%. The spectral parameters for the same overlapped region
from two adjacent window calculations are similar but slightly different, especially
for those that describe noise. Instead of trying to match two lists to remove the
redundancy, which could be very tricky and unreliable, a simplified approach is used
here: for each window calculations, only the parameters for describing the poles that
locate inside the center half of the spectral window is retained. By concatenating
parameters from all window, we can obtain an approximate “complete” line list for
the whole spectral range. Figure 2.27 compares the signal reconstructed using such
a line list to the original one. It is not surprising that the initial part of the signal,
which is mostly defined by the broad features, is not reproduced well, due to the lack
of entries for the broad features. The rest of the signal is reproduced very well. The
difference between the original and reconstructed signals is about the noise level. This
proves that most, if not all, the narrow features are parametrized quite accurately.
However, this observation should not be extended to the conclusion that all the poles
computed by FDM are genuine. It is highly possible that multiple Lorentzian lines
are responsible for describing a single genuine peak, especially in the case of noisy
signals with heavily overlapped features and/or lineshape distortions.
149
2.6.2 Possible Ways for Checking Individual Peaks
Figure 2.27 is an example where we check the overall validity of the parametric
fit provided by FDM. In many cases, we are also interested in knowing whether a
particular entry in the FDM line list is genuine or not, and if yes, what the uncertainty
(error bar) is. These are very difficult problems where the ultimate solutions are still
to be found. However, there are a few qualitative and semiquantitative methods for
estimating the error bar and reliability of individuals peak computed by FDM.
Method 1: There existed a method for estimating the error of each eigenvalue since
FDM was first introduced [24, 67], which was simply based on
ε(n) = ||(U (n) − unkU
(0))Bk||, n ≥ 2 ,
where U (n) is the matrix representation of n-step evolution operator Un
= exp(−inτ Ω).
Typically n=2 is used. The spurious poles usually have a much larger estimated error
and this had been used to filter out spurious poles in the early days of FDM. It was
later discovered that it should only be used as a qualitative estimation, instead of a
rigorous criteria. Broader features, genuine as well as spurious, tend to have larger
estimated errors, and some narrow spurious poles might have small estimated errors
by accident. However, it is relatively safe to treat those narrow poles with large
estimated errors as spurious poles.
Method 2: This method relies on estimating the “velocity” of eigenvalues numer-
ically by small random noise perturbation. It is assumed that eigenvalues corre-
sponding to genuine poles tend to be less sensitive to small perturbations while the
spurious eigenvalues behave oppositely. Numerically, one can first run a FDM cal-
150
culation using the original signal, then add small pseudo-random noise perturbation
(∼ 1%) to the signal and re-run the FDM calculation. Then one needs to match
up two sets of eigenvalues and estimate the velocity of eigenvalues numerically by
v(ωk) ∼ δωk/|noise|. The eigenvalue matching step might be tricky, but numerically
experiments showed that it could usually be achieved as long as the noise perturbation
was sufficiently small. The velocity estimation is typically consistent with the error
estimation described in previous paragraph. This estimation is also only qualitatively
correct. There is no guarantee that all spurious peaks can be detected, nor that all
eigenvalues with large velocities are spurious.
Method 3: This method involves injecting a narrow Lorentzian pole close to the
peak(s) of interest, rerunning the FDM calculation and comparing the results. This
is also based on the assumption that genuine peaks are more stable than the spurious
ones. If the peak is genuine, it is unlikely that injecting a small Lorentzian pole close-
by will have a large effect on it. However, if the peak corresponds to some poorly
converged features or noise, any change of the close-by environment might significantly
change the local results. A systematic scheme for carrying out this “probe-and-test”
procedure requires more studies on what kind of probe pole (e.g., peak intensity and
linewidth) should be used and where to inject it.
Method 4: Finally, one can always study the convergence of a particular peak as
a function of FDM/RRT processing parameters such as basis size, basis density, sig-
nal length used, and regularization level (for RRT). If a feature is stable or becomes
stable with respect to the change of these parameters, we can have more confidence
saying that it is a genuine peak. This procedure is more time consuming and might
151
seem unnecessary in 1D case. However, in 2D or higher dimensional applications, it
is almost always necessary to run the FDM/RRT calculations multiple times, par-
ticularly using various signal sizes, before one can be confident that the results have
successfully converged.
2.6.3 A Semi-Quantitative Way for Estimating the Sensitiv-
ity of FDM/RRT Spectral Estimations
Here we describe a semi-quantitative way for estimating the sensitivity of FDM/RRT
spectral estimations. The idea is very simple. The spectral estimation I(ω) com-
puted by FDM/RRT has a periodic dependence on the spectral window position. If
one computes the spectral intensity at a particular frequency ω0 multiple times us-
ing different spectral window conditions, the results will differ, even though only by
small amount. The degree of variance provides us a semi-quantitative way of measur-
ing the sensitivity (or uncertainty) of FDM/RRT spectral estimation. For example,
Figure 2.28 illustrates this uncertainty using a special setup called Continuous Win-
dowing FDM/RRT, or CW-FDM/RRT. In CW-FDM/RRT, for each frequency point
ω, a spectral window centered at ω is first set up and the U matrices constructed.
Then these matrices are either diagonalized or used in RRT to compute the spectral
intensity only at ω. Same procedure is repeated for each frequency grid. Obviously
this is more time consuming than the normal FDM/RRT algorithm where the same
U matrices are used to compute spectral estimation for all frequency points inside the
spectral window. However, this setup reveals the uncertainty of FDM/RRT spectral
152
100 110 120 130 140 150 160
150
150
Frequency (Hz)
Continuous Windowing FDM
FDM with single window
x 150
x 150
Figure 2.28: Sensitivity of FDM/RRT calculations illustrated by using the continu-
ous windowing FDM. While the baseline of normal FDM calculation is perfectly flat,it is highly oscillative in CW-FDM spectrum. The amplitude of the oscillation pro-vides useful information about the sensitivity (or uncertainty) of FDM/RRT spectralestimations.
estimations that is hidden in the normal FDM/RRT calculations. For example, while
the baseline of the bottom trace of Figure 2.28 is flat, highly oscillative behaviour is
observed in the top trace, computed by CW-FDM (CW-RRT spectrum is similar and
not shown). The period of the oscillation is purely determined bye the signal length,
but its amplitude depends on all the FDM/RRT parameters such as the basis density,
basis size and regularization level (for RRT). It is possible to use the amplitude of
this oscillation as a semi-quantitative measurement of the uncertainty of FDM/RRT
spectral estimations. One can simply use the amplitude of the oscillation as the error
153
20 40 60 80 100 1200
0.5
1
1.5
2
2.5
3
Sensitivity of FDM/RRT vs. Basis Size
Basis Size
Sen
siti
vit
y (
A.U
.)
Square: a model signalTriangle: an experimental signalCircle: an experimental signal
0.6 0.8 1 1.2 1.40
10
20
30
40
50
Sensitivity of FDM/RRT vs. Basis Density
Basis Density
Sen
siti
vit
y (
A.
U.)
Square: a model signalTriangle: an experimental signalCircle: an experimental signal
Figure 2.29: Sensitivity of FDM/RRT spectral estimation as functions of basis sizeand basis density using two experimental NMR signal and a model signal with inte-gerization noise.
bar for the spectral estimation. Another, more interesting application is to use it
to study the sensitivity as a function of all FDM/RRT parameters and find out the
optimal processing parameters for typical signals. For example, Figure 2.29 plots the
sensitivity of FDM/RRT spectral estimations as functions of the basis size and basis
density using two experimental NMR signals and a model signal with integerization
noise. We can see that basis size of Nb = 50 and basis density of ρ = 1.1 are the
most efficient values for obtaining stable FDM/RRT spectral estimations for these
three signals. Previous numerical experiments also suggest that these are indeed the
most efficient optimal conditions for processing typical 1D signals. It is very interest-
ing that such a semi-quantitative method is able to re-predict the optimal processing
parameters in a straightforward, explicit way.
154
2.6.4 Reliability of 2D FDM
When 2D FDM and RRT are used as spectral estimators, one can also use inverse
FT to recover the time domain signal and compare to the original one. However,
this method is less convenient in 2D or higher dimensional cases due to the following
reasons: first, to compare the signals in the time domain, spectra over the whole
spectral range must be computed, while we are typically only interested in a few
local regions; second, multi-dimensional NMR signals typically contain a water signal,
which is often much more significant than all the other signals. Thus it is suboptimal
to include the water signal in checking the fitting quality of the small features of
interest. Instead, it is more efficient and more reliable to compare the results with
the FT of the original signal locally in the frequency domain.
The infinite 2D FT spectrum is typically estimated by 2D FDM. In order to make
direct comparison, we need to estimate the finite 2D FT spectrum using the same
spectral parameters. This can be done by replacing the double-absorption expression
of Eq. 2.68 with the following expression, obtained the analytically evaluating the
finite 2D FT summation17
AN1,N2(ω1, ω2) ≈ τ1τ2
∑
k,k′
Re[Dk,k′] (2.110)
Re
[
1 − eiN1τ1(ω1−ω1k)
1 − eiτ1(ω1−ω1k)− 1
2
]
Re
[
1 − eiN2τ2(ω2−ω2k′ )
1 − eiτ2(ω2−ω2k′ )− 1
2
]
.
By computing the difference of the FDM estimation of 2D DFT with the actual
2D DFT of original signal, one can have a rough measurement of the fitting quality.
17Note that FT with certain apodizations such as exponential and trigonometric functions can beevaluated analytically and incorporated.
155
-800 -600 -400 -200 0 200 400 600 800
800
600
400
200
0
-200
-400
-600
-800
800
600
400
200
0
-200
-400
-600
-800
800
600
400
200
0
-200
-400
-600
-800
-600 -400 -200 0 200 400 600 800
2D FT with N1xN2=200 x642D FT with N1xN2=200 x16
Estimated infinite time 2D FT
Estimated 2D FT with N1xN2=200 x64Estimated 2D FT with N1xN2=200 x16
a) b)
e)
d)c)
f)
Difference of spectra b) and d)
1H / Hz 1H / Hz
15N
/ Hz
15N
/ Hz
15N
/ Hz
Figure 2.30: An example where the spectral parameters obtained by FDM are used toestimate finite 2D FT spectra for direct comparison with the actual 2D FT spectra.The signal used is the same 1H-15N HSQC signal of rubredoxin used in previous2D examples. The FDM spectral parameters were obtained using 200×16 N-typecomplex data points. The actual 2D FT spectra (panel a and b) were obtained usingboth N- and P-type data sets with cosine weighting functions in both dimensions. Itshows that the estimated finite 2D FT spectra match with the actual 2D FT spectravery well. The mismatch between spectra (a) and (c) is only about 1.9% (see text).Even when the same parameters obtained using only 200×16 data points were used toestimate the finite 2D FT with length of N1 ×N2 = 200× 64, the mismatch betweenspectra (b) and (d) is only 2.9%. Note that the noise that is only present in long 2DFT spectra (panel b) can be accounted for a large percentage of the larger mismatch.
156
Figure 2.30 is a demonstration of such a checking procedure using the same 2D HSQC
NMR experimental signal of rubredoxin used in previous 2D examples. Actual 2D FT
spectra were computed by using both N- and P-type data set with cosine weighting
functions in both dimensions. Both a short 2D FT spectrum (panel a, 200× 16) and
a long one (panel b, 200 × 64) are shown. Note that the long 2D FT spectrum is
noisier. Spectral parameters ω1k, ω2k′, Dk,k′ were computed by FDM2K only using
200 × 16 complex data points. These parameters were then used to estimated two
finite time 2D FT spectra with signal length N1 × N2 being 200 × 16 (panel c) and
200 × 64 (panel d) respectively18, and the infinite time 2D FT spectrum (panel e).
One can simply stare at these spectra (such as panel a and c) and see that they match
very well. Quantitatively, one can compute a relative “matching error” based on,
ε =
∑ |AN1,N2(ω1, ω2) − FTN1,N2(ω1, ω2)|2∑ |FTN1,N2(ω1, ω2)|2
, (2.111)
where FTN1,N2(ω1, ω2) denotes the actual 2D FT spectrum. The as calculated match-
ing error between spectra (a) and (c) is only 1.9%, which is within the noise level.
This is might not be surprising as we did use 200 × 16 data points to compute these
parameters in FDM and a good match is expected. We can also use the same spectral
parameters to estimate a long finite time 2D FT spectra with N1 × N2 = 200 × 64,
four times longer than the signal size used in FDM calculation. Again, the estimated
2D FT spectrum (panel d) matches with the actual 2D FT (panel b) very well, with
the matching error being only slightly larger, 2.9%. Note that a significant portion
of the increased mismatch can be due to the noise that is only present in the long 2D
18Note that cosine weighting function is evaluated analytically, resulting a formula that is slightlydifferent from Eq. 2.110.
157
FT spectrum. This provides a very strong evidence that FDM is capable of providing
a good, reliable fit of the time signal to the Lorentzian model.
When the signal does not satisfy the Lorentzian model as well, the fitting error
will be larger. More examples will be given in Chapter 3, where FDM2K is applied to
analyzing 2D and 3D protein NMR signals. It has been shown that FDM is capable of
providing reliable fit of both large and complex multidimensional NMR signals in most
cases. The reliability and accuracy of individual peaks can be also be estimated, using
the techniques listed in previous section. In practice, these approximate error bars do
not always provide much information about the authenticity of the peak. Instead, it
is more useful and necessary to study the behaviour of particular peak(s) as a function
of processing parameters, especially the signal length used. More discussions can be
found in Chapter 3.
2.7 Summary and Remarks
Two nonlinear methods, namely the Filter Diagonalization Method (FDM) and
Regularized Resolvent Transform (RRT), for high-resolution spectral analysis of time
domain signals have been presented.
FDM solves the highly nonlinear fitting problem of Harmonic Inversion Prob-
lem (HIP) by pure linear algebra of diagonalizing some small data matrices in the
frequency domain. FDM is intrinsically stable and efficient due to its two most im-
portant properties: solving the nonlinear fitting problem via linear algebra and local
spectral analysis via Fourier-type basis. Applied to both model and experimental
158
signals, it has been shown that FDM is able to deliver resolution beyond the FT
time-frequency uncertainty principle, given that the signal sufficiently satisfies the
Lorentzian model and is long enough (but typically not long enough for FT to fully
resolve all the features). The actual power of FDM lies in its multi-dimensional ex-
tensions (MD-FDM). MD-FDM is a true multi-dimensional spectral analysis method,
which is able to process the whole data set to characterize the multi-dimensional fea-
tures. The achievable resolution in all dimensions is determined together by the
total information content of the signal, instead of the signal sizes along individual
dimensions. MD-FDM is ill-conditioned due to several fundamental and practical
limitations. Regularization techniques must be used in order to obtain stable and
meaningful results with single FDM calculations. Among all the regularization meth-
ods exploited, FDM2K and SVD are particularly efficient. However, the ultimate
optimal regularization is yet to be found. In conclusion, 1D FDM is basically a well-
developed method, that is stable, efficient and capable of provide resolution higher
than FT spectral analysis. Stable, high-resolution multi-dimensional spectral estima-
tions can be obtained efficiently by MD-FDM with regularizations, but the problems
associated with processing multi-dimensional signals have only been partially solved.
Even though the results we have obtained are already superior compared to those
given by FT or other high-resolution methods such as LP and MaxEnt, there is still
plenty of room for further improvements.
RRT emerges naturally from FDM. The same data matrices are used but spectral
estimations are directly computed without the diagonalization step. Implementation
of RRT is straightforward and numerically efficient. Conventional techniques such as
159
Tikhonov regularization or SVD can be efficiently use to regularize RRT. RRT can be
also use to construct non-FT types of spectral representations such as inverse Laplace
transform and various pseudo-absorption spectra. The advantage of RRT lies in its
simplicity and transparency, and the drawback is the lack of the freedom to directly
manipulate individual peaks. Both 1D and MD RRT are well-developed methods
that are stable, reliable and efficient. The problems associated with RRT are mostly
understood and significant improvements over current version of RRT are not much
expected. The major limitation of RRT lies in the phase instability of the peaks,
manifesting it as the lineshape distortions in the final absorption spectra, especially
when the signal is truncated in all dimensions. While the hybrid method eXtended
Fourier Transform (XFT) intends to combine the stability of FT with high-resolving
power of RRT, the results are not totally satisfactory. In high-resolution XFT spectra
where the contribution from RRT correction is significant, the phase instability is still
present.
Finally, a limitation shared by RRT and FDM is the lack of a consistent way for
verifying the results. While it is possible to qualitatively check whether a particular
feature is likely to be genuine or not, there is not a rigorous, quantitative way to
make some solid, conclusive judgments. False positives as well as false negatives can
be costly for those unaware of the underlying uncertainty. One of the main focuses
for future research should be developing some systematic schemes for detecting false
positives and false negatives. Before such a scheme becomes available, the question
of “I see something great, but is it real?” will always be haunting us and hamper
FDM/RRT from being widely accepted and used.
160
Appendix I: Non-Hermitian Quantum Mechanics
Here we briefly review the linear algebra of Non-Hermitian Quantum Mechan-
ics [91]. The conventional quantum mechanics considers only Hermitian operators
acting in a Hilbert space. The Hilbert space is characterized by a Hermitian inner
product 〈Ψ|Φ〉 = 〈Φ|Ψ〉∗ with the asterisk defining the complex conjugate. Any vec-
tor in Hilbert space has is a well defined the norm, 〈Ψ|Ψ〉, which is always real and
non-negative. The eigenvalues of a Hermitian operator are always real. However,
Hermitian quantum mechanics is not sufficient to describe dissipative dynamic sys-
tems, as we need non-Hermitian operators with complex eigenvalues representing the
frequencies of decaying sinusoids.
Consider an abstract non-Hilbert linear vector space A. To distinguish between
the Hermitian and complex symmetric inner product for the latter we use the round
brackets, (Ψ|Φ) = (Φ|Ψ). Note that (Ψ|Ψ) is not necessarily real, i.e., the norm of Ψ is
not necessarily defined in our non-Hilbert space. Moreover, (Ψ|Ψ) can even vanish for
a nonzero vector Ψ. Although numerically this is unlikely to happen, it is clearly an
indication of possible problems (e.g., instability) in the numerical algorithms involving
the non-Hermitian inner products.
We will always identify linear operators, U , Ω, etc., that act on vectors in A by
a cap. By the complex symmetric operator Ω we mean that it satisfies the following
relationships:
(Ψ|Ω|Φ) = (Ψ|Ω|Φ) = (Ψ|Ω|Φ) , (2.112)
for any two vectors |Ψ) and |Φ) from A. In words, it does not matter whether we first
161
operate with Ω on |Φ) and then evaluate the inner product with (Ψ| or vice versa.
An operator Ω is diagonalizable, if it has a set of eigenvalues ωk and eigenvectors
|ωk) satisfying
Ω|ωk) = ωk|ωk) , (2.113)
where the eigenvectors are orthonormalized with respect to the complex symmetric
inner product, i.e.,
(ωk|ωk′) = δkk′ . (2.114)
We will also assume implicitly that our operators are not pathological, which, in
particular, means that the eigenvectors form a complete basis and one can use the
resolution of identity,
I =∑
k
|ωk)(ωk| . (2.115)
This also implies that Ω can be expressed using the spectral representation,
Ω =∑
k
ωk|ωk)(ωk| . (2.116)
The spectral representation becomes very useful when we want to obtain an ex-
pression for a function f(Ω) of an operator Ω, whose eigenvalues and eigenvectors are
known:
f(Ω) =∑
k
f(ωk)|ωk)(ωk| . (2.117)
Note that f(Ω) is also an operator with the eigenvalues fk = f(ωk) and the same
eigenvectors |ωk).
162
Appendix II: An alternative expression for comput-
ing the amplitudes
Here we derive an alternative expression for computing the amplitudes, which is
more accurate for narrow poles. As for normal eigenvalue solvers, the eigenvalues are
obtained variationally, while the eigenvectors are obtained in a non-variational way.
Thus according to Eq. 2.24 the amplitudes dk are generally much less accurate than
the frequencies ωk. In addition, given an eigenvector Bk, Eq. 2.24 only makes use
of the first half (M = N/2) of the time signal available and is not always the most
accurate expression for the coefficient dk that exists, especially for a narrow pole ωk.
As such a more accurate formula is derived for the narrow poles. First let us rewrite
Eq. 2.6 in a more general form:
d1/2k ≡ (Υk|Φ0) =
(
[
U/uk
]n′
Υk|Φ0
)
= u−n′
k
(
Υk|Un′
Φ0
)
= u−n′
k
∑
j
[
Bk
]
j(Ψj|Φn′)
= en′γ∑
j
[
Bk
]
j
M−1∑
n=0
ein′τ(ωk+iγ)einτϕjcn+n′ ,
where γ and n′ are free parameters. We can now average the above expression over
n′ = 0, 1, . . . , Maver − 1 for an arbitrary Maver between 1 and M ,
d1/2k =
1 − e−τγ
1 − e−Maverτγ
∑
j
[
Bk
]
j
×Maver−1∑
n′=0
M−1∑
n=0
ein′τ(ωk+iγ)einτϕjcn+n′ , (2.118)
where the averaging was done by weighting each term with e−n′γ (to eliminate the
prefactor) and then normalizing the final result by[
∑Maver−1n′=0 e−n′γ
]−1. Just like in
163
Eq. 2.18 one of the two sums in Eq. 2.118 can be evaluated analytically (see Section
2.2.4). To eliminate the ambiguity in the choice of the free parameters in Eq. 2.118 for
narrow poles ωk (for which Eq. 2.118 is relevant), it suffices to stick with Maver = M
and
γ =
−Imωk , Imωk < 0
0 , Imωk > 0
.
For model signal or real signals with very high SNR, Eq. 2.118 does yield the ampli-
tudes with more accuracy. However, it will lead to an ambiguous result if Mτ |Imωk|
1, in which case the signal corresponding to these “broad” peaks should decay away
very quickly and thus including longer signal to calculate their amplitudes would only
deteriorate the accuracy. In these cases, Eq. 2.24 should be used. A final word on
whether Eq. 2.24 or Eq. 2.118 should be used is as following: unless the signal has very
high SNR and one is really interested in computing the amplitudes of some narrow
poles as accurately as possible, Eq. 2.24 should always be used.
164
Appendix III: Other Fourier-Type Basis
The rectangular-window Fourier basis defined in Eq. 2.15 is very simple and easy
to implement. However, experience with Fourier Transform spectral analysis might
suggest that it might not the most efficient one, for example, considering the highly
oscillative sinc-like structure of off-diagonal matrix elements. Here we discuss a more
general definition of Fourier basis by including a filter function gn,
|Ψ(ϕj)) =M−1∑
n=0
gn einτϕj |Φn) . (2.119)
Mathematically, any non-vanishing filter functions gn are exactly identical in the
full space. However, the point of using a Fourier-type localized basis is to reduce
the problem to a small subspace. It is possible, but not obvious, that a good filter
function might lead to more accurate and stable numerical approach.
In general, the efficient expressions we derived for computing the U matrix ele-
ments (Eq. 2.20 and Eq. 2.23) can not be used. Chen and Guo [62] have derived an
alternative expression to compute the U matrix elements in a Fourier basis with a
general filter. This expression requires Krmwin times more CPU time compared to
the simple case of rectangular-window Fourier basis of Eq. 2.15. However, there are
several other type of filter functions that can be implemented with similar compu-
tational efficiency of rectangular window function [67]. For example, a exponential
filter function,
gn = exp(−γn) , (2.120)
corresponds to replacing the real frequency grid ϕj by a complex one, ϕj + iγ, and
expressions Eq. 2.20 and Eq. 2.23 can still be used.
165
Trigonometric type of filter functions can also be efficiently implemented. For
example, a cosine filter function,
gn = cos(
π
2
n
M + 1
)
, (2.121)
is often used in Fourier Transform and leads to a good balance between the suppression-
of sinc wiggles and efficient sampling of initial part of the signals. In this case,
Eq. 2.119 can be rewritten as in terms of rectangular-window Fourier basis,
|Ψ(ϕj))cos =1
2( |Ψ(ϕj + α)) + |Ψ(ϕj − α)) ) , (2.122)
where α = π/(2(M + 1)). Then the U matrix elements can be computed using the
same expressions, Eq. 2.20 and Eq. 2.23.
To our disappointment, numerical experiments showed that introducing these win-
dow functions had not led to a significant numerical difference. This indicates that
a rectangular-window Fourier basis is already very efficient in carrying out the local
spectral analysis. The use of different filter functions only affect the initial structure
of U matrices and have minor effect on the whole subspace formed by Kwin window
basis functions.
166
Chapter 3
Application of FDM for NMR
Spectral Analysis
3.1 Introduction
Nuclear Magnetic Resonance (NMR) spectroscopy is one of the most powerful
tools for probing the structure of the physical matters. It is one of most commonly
used non-destructive analytical tools for studying the structure of chemical com-
pounds; it is one of the only two methods for determining the 3D structure of biolog-
ical macromolecules, and the only one that works in solution. In addition, NMR is
a powerful tool for studying the dynamics and interactions between molecules, pro-
viding unique information for understanding how biomolecules such as proteins and
DNA function in their natural environments.
NMR is based on the principle of nuclear spin angular momentum, an intrinsically
quantum mechanical property that is analogous to electron spin angular momentum.
167
It was first detected in bulk matter by Bloch [104] and Purcell [105] in 1946, who
then shared the Nobel Prize for Physics in 1952 for this discovery. In the molecule,
the nuclei are surrounded by electrons, which act like a shield to the external mag-
netic field. Thus the actual strength of magnetic field experienced by the nucleus
has dependence on the local electronic environment. The observed nuclear magnetic
resonance frequencies will be slightly different for nuclei in different chemical envi-
ronments. This frequency shift is thus called the “chemical shift”. The chemical shift
was first predicted by Dharmatti and then demonstrated by the famous three-line
proton NMR spectrum of pure ethanol [106]. Even though the existence of chemical
shift is a big disappointment for physicists [107], it provides a way for chemists to
study the structure of chemical compounds. At that time, NMR spectroscopy was
called continuous wave NMR (CW-NMR) as it relied on sweeping either the magnetic
field or the radio-frequency to detect difference resonances. Due to its low sensitivity
and long experimental time, NMR did not actually gain much popularity in organic
chemistry, until Ernst and Anderson developed modern Fourier-Transform NMR (FT-
NMR) [108] in 1966. Short bursts of RF field, or pulses, were applied perpendicular
to the static magnetic field to excite all the resonances at once. A Free Induction
Decay (FID) time domain signal was then acquired and Fourier Transformed to pro-
vide the frequency spectrum. FT-NMR speeded up the experiment by hundreds of
times and improved the signal to noise ratio (SNR) enormously for the data obtained
in the same amount of time. With the development of the Fast Fourier Transform
(FFT) algorithm by Cooley and Tukey [6] and decreasing cost of computers in the
late 1960s, FT-NMR grew explosively and almost completely replaced the CW-NMR.
168
Another milestone in the history of NMR spectroscopy was the introduction of
two-dimensional NMR experiments by Ernst [109] in 1976, based on the idea proposed
by Jeener [110] in 1971. By introducing a second dimension, not only could the
potentially heavily overlapped resonances be dispersed into the second dimension, but
useful information about the chemical and/or dynamical connections between spins
could be obtained. With the aid of a quantum-mechanical representation known as
the density matrix formalism [111, 112], which provides a complete description of the
state and dynamics of the spin system, numerous novel 2D and higher dimensional
NMR experiments were developed. In 1991, Ernst was warded the Nobel Prize for
Chemistry for his pioneering work on modern NMR methodology. Nowadays, 1D and
multidimensional NMR experiments are performed on a routine basis in many fields
including chemistry, solid state physics, molecular biology, and medicine [113, 11].
The effectiveness of NMR hinges on the ability to obtain a resolved spectrum.
While sensitivity of NMR has been steadily improved by new advances in NMR
probe and high-field magnet technologies over the last two decades, the fundamental
resolution of NMR solely depends on the obtainable magnetic field strength and has
only seen limited improvement [114]. The resolution that can actually be achieved
is further limited due several severe drawbacks of conventional FT spectral analy-
sis. In particular, the FT resolution is limited by the so-called FT time-frequency
uncertainty principle, which means a long time signal has to be acquired in order to
obtain a high spectral resolution. What makes it worse is that the FT uncertainty
principle applies to all dimensions independently in multidimensional FT spectral
analysis. Due to various practical limitations, multi-dimensional NMR signals are
169
typically truncated or severely truncated in the indirect time dimensions, leading to
poor FT resolution in the corresponding frequency dimensions. This is one of the
major limitations of the multi-dimensional NMR experiments. As we discussed in
Chapter 1, considerable efforts have been spent on developing various high-resolution
alternatives to FT spectral analysis. However, due the high complexity and large
scale of the problem, these methods are typically both computationally expensive
and numerically unstable. Accordingly, FT remains the method of choice in most
cases for NMR spectral analysis.
The FDM and RRT algorithms introduced in Chapter 2 have proved to be superior
in computational efficiency and numerical stability. It shows promise as a working
method for high-resolution spectral analysis. Multi-dimensional FDM and RRT are
true multidimensional methods. The whole data set are processed together to provide
resolution enhancements in all dimensions, according to the total information content
of the signal. FDM and RRT are thus particularly suitable for NMR spectral analysis.
In this chapter, we will discuss various issues that are specific to analyzing NMR
experimental signals. The rest of the chapter is organized as following: we will first
discuss the linear phase correction via FDM/RRT for 1D NMR signals in Section 3.2,
then in Section 3.3, we will discuss the construction of non-trivial projections and its
application to a singlet-TOCSY experiment. In Section 3.4, an efficient scheme for
processing multi-dimensional Constant-Time (CT) NMR signals is introduced and
then applied to 2D CT-HSQC and 3D CT-HNCO NMR experiments of proteins.
Finally, a summary is given in Section 3.5.
170
3.2 Application of FDM/RRT for 1D NMR
From the spectral analysis point of view, 1D NMR does not really have limi-
tations, simply because long signals are typically available and high resolution FT
spectra can be thus obtained. The resolution enhancement that can be achieved is
marginal, even for methods like FDM or RRT. However, FDM and RRT can pro-
vide additional useful information besides the high-resolution spectra. For example,
as discussed in Section 2.5.3, RRT can be used to generate a pseudo-2D spectrum
I(ω + iΓ) as a function of both real and imaginary frequencies and separate poten-
tially heavily overlapped resonances. FDM can provide a parameter list which is
generally a reasonable fit of the time signal, as was demonstrated in Section 2.6.1.
Even though multiple entries might be used to represent a single spectral feature and
any nontrivial use of line list should be done with cautions, the line list provides a
much better starting point for peak picking, peak integration and coupling constant
measurement, while the conventional peak searching methods are sensitive to noise,
baseline distortion, and closeby strong features. In this section, we will focus on cor-
recting the linear phase by FDM and RRT, where conventional FT spectral analysis
has problems.
3.2.1 Linear Phase Correction
Problems with the first few data points of NMR signal can arise from many sources
including the transient response of the audio filter and probe ring-down. Linear
phase delay can be also caused by the stabilization delay following the magnetic field
171
gradient pulses and some special pulse sequences such as 1-3-3-1 water suppression
sequence [115]. In these cases, the signal c(t) is only available from some time t0 > 0
instead of t0 = 0. In 1D NMR, all resonances are typically controlled to have the
same phase at t = 0, giving rise to a nice absorption spectrum with all peaks “in
phase”. It is thus desirable to produce such a “in phase” spectrum even when the
signal does not start from t = 0, requiring a time shifting or so called “linear phase
correction” in FT,
I(ω) =∫ T
t0C(t)eiωtdt = eiωt0
∫ T−t0
0c(t′)eiωt′dt′ = eiωt0 FT[c(t′)] , (3.1)
where the term eiωt0 corresponds to a phase correction that is linear with respect to
the frequency, ω. The FT linear phase correction works well only when the delay
is short, so that the phase roll over the typical linewidth is small. A large linear
phase correction can cause severe phase rolls in the baseline. For example, trace (c)
of Figure 3.1 shows that, with the delay t0 = 17ms, the baseline is severely distorted
after the FT linear phase correction. The source of the baseline distortion is follows:
FT corrects the linear phase point by point on the frequency axis, which is only
accurate at the center of the peaks. In another word, the phase roll is due to the fact
that FT is not able to recognize individual peaks and correct their phases accordingly.
For example, considering a signal that contains an isolated resonance at zero frequency
with a linewidth of 5 Hz, and assuming a delay of 10 ms, the phase corrections for
spectral points at -5 Hz and +5 Hz differ by ∆φ = 10 × 0.010 × 360 = 36o, while
the phase correction should be the same as across the same peak (at zero frequency).
FDM and RRT fit the signal to a sum of Lorentzian lines and thus can potentially
172
-4000 -3500 -3000 -2500
Frequency (Hz)
a) No delay, 1D RRT
c) 17 ms delay 1D FDM via Eq. 3.1
b). 17 ms delay DFT, phase corrected
d) 17 ms delay 1D RRT via Eq. 3.4
Figure 3.1: Linear phase correction via DFT and FDM/RRT, applied to a NMRexperimental 1D signal with 5K complex data points. The FT linear correction intro-duces severe phase rolls in the baseline (trace b). The FDM linear phase correctionvia Eq. 3.2 is numerically unstable and can also introduce baseline distortions (tracec). By incorporating the linear phase correction into the RRT expression or thegeneralized eigenvalue problem, the numerical instabilities introduced by Eq. 3.2 canbe eliminated (trace d). However, there are still some small instabilities due to theexistence of broad poles and noise, which can be further suppressed by FDM/RRTaveraging.
recognize all peaks and correct the phases of all peaks in a more consistent way.
In FDM is used, we have all the spectral parameters ωk, dk(t0), where dk(t0)
denotes the complex amplitude at t = t0 > 0. They can be then used to estimate the
amplitudes at t = 0 simply by
dk(0) = dk(t0) exp(iωkt0) . (3.2)
However, such a correction is only reliable if both the frequencies ωk and the ampli-
173
tudes dk(t0) are accurately extracted. This is often true for the narrow peaks when
the signal sufficiently satisfies the Lorentzian model (as we demonstrated in Sec-
tion 2.6.1). However, for broad features, the parameters extracted are less accurate,
and such a correction might introduce a large error and result in baseline distortions.
In addition, multiple entries might often be responsible for describe a single spectral
feature. The mutual cancellation and interference might not be conserved after the
linear phase correction, resulting in distorted lineshapes. Finally, the line list con-
tains entries that describe noise. Linear phase corrections for these entries do not
make sense and might amplify the noise. Unfortunately, there is no reliable method
for separating noise spikes from genuine features. As a result, Eq. 3.2 can introduce
numerical instabilities. For example, trace (b) of Figure 3.1 shows an example where
Eq. 3.2 is directly used to correct the linear phase caused by a delay of 17ms for an
experimental signal that contains noise and broad features. Even though it seems that
the correction is quite good for narrow peaks and there is no phase roll in the base-
line, there is a big baseline distortion due to some broad poles around ω ∼ −2750Hz.
An averaging method that is similar the stabilization scheme proposed by Mandelsh-
tam [116] can be used to suppress the instabilities. The idea simply relies on the fact
that the errors in the poles are random and thus can be reduced by averaging over
many FDM calculations with varying signal length. A successful example was given
in Ref. [67].
Here we present an alternative scheme where the linear phase correction is in-
corporated into the generalized eigenvalue problem and can significantly reduce the
instabilities. RRT expressions can also be used to obtain the linear phase corrected
174
spectral estimation directly from the signal in this scheme.
Given a discrete time signal c(n) ≡ c(nτ), available only from n = n0, n0 +
1, . . . , n0 + N − 1, the goal is to estimated the infinite time DFT summation1,
I(ω) = τ∞∑
n=0
c(n)eiωnτ . (3.3)
Substituting the FDM quantum ansatz, c(n) =(
Φ0|UnΦ0
)
, into Eq. 3.3 and evaluat-
ing the geometric summation analytically leads to2,
I(ω) = τ
(
Φ0
∣
∣
∣
∣
∣
1
1 − eiτωU
∣
∣
∣
∣
∣
Φ0
)
= τ
(
Φn0
∣
∣
∣
∣
∣
U−n01
1 − eiτω UU−n0
∣
∣
∣
∣
∣
Φn0
)
= τ
(
Φn0
∣
∣
∣
∣
∣
1
U2n0 − eiτω U2n0+1
∣
∣
∣
∣
∣
Φn0
)
. (3.4)
Evaluated in the Fourier basis defined in Eq. 2.15, the operator expression of Eq. 3.4
becomes a numerical expression for estimating the linear phase corrected spectra
directly using the delayed finite time signal:
I(ω) = τ CT
n0
[
U(2n0) − eiτωU
(2n0+1)]−1
Cn0 = τ CT
n0R
(2n0)(ω)−1 Cn0 , (3.5)
where the shifted FT column vector Cn0 is computed as,
[Cn0]j = (Φn0 |Ψj) =M−1∑
n=0
einτϕjc(n + n0) ,
and the delayed evolution matrices U(2n0+l), l = 1, 2 can be computed using Eq. 2.20
and Eq. 2.23 with p = 2n0 and p = 2n0 + 1 respectively. Note that M = (N − n0)/2
due to the condition that 2(M − 1) + 2n0 + 1 = N + n0 − 1.
1The zero point correction term is dropped here for the sake of simplicity.2Assume that for dissipative systems, U∞ → 0.
175
Obviously, Eq. 3.5 can be directly evaluated by computing a pseudo-inverse of the
resolvent matrix via either SVD or Tikhonov Regularization. Alternatively, we can
solve a generalized eigenvalue problem,
U(2n0+1)
Bk = ukU(2n0)
Bk , (3.6)
compute the amplitude (already linear phase corrected) as
d1/2k (0) = C
T
n0Bk , (3.7)
and then use Eq. 2.27 to construct the linear phase corrected FDM ersatz spectrum.
In principle, both approaches should give essentially the same result. However, as
a large linear phase correction is ill-conditioned for signals with noise, broad poles
and/or heavily overlapped features, the RRT approach is more suitable for obtaining
a stable spectral estimation.
By incorporating the linear phase correction into the RRT expression or general-
ization eigenvalue problem, we totally avoid the possibility of amplifying the numerical
errors in the spectral parameters by using a separate linear phase correction step of
Eq. 3.2. Expression Eq. 3.5 is numerically much more stable and can potentially
used to correct much larger time delays. For example, Figure 3.2 demonstrates the
robustness of Eq. 3.5 using the model signal “Jacob’s Ladder” that has been used in
previous 1D examples. It shows that Eq. 3.5 can be used to apply the linear phase
correction for virtually any length of delays without running into numerical insta-
bility problems. Applied to the experimental signal, Eq. 3.5 also shows significant
improvement in baseline stability compared to both DFT and Eq. 3.2, demonstrated
176
3 3.5 4 4.5
Nsig = 32 K
Frequency (Hz)
Delay = 0
Delay = 0.1 s
Delay = 1.0 s
Delay = 10 s
a.
b.
c.
d.
Figure 3.2: A demonstration of the robustness of linear phase correction via Eq. 3.5using the model signal “Jacob’s Ladder”. Shown here is a dense region of the spec-trum. Only 32 K data points were used and thus the multiplets were not totallyconverged, even though sufficiently resolved. Trace (b) through (d) show the spectraobtained by first skipping 100 up to 10000 data points before reading in 32K datapoints from the rest of the signal, corresponding to time delays of t0 = 0.1 second upto t0 = 10 seconds. Even with a delay of 10 seconds, a reasonably good spectrum canstill be obtained. However, note that small artifacts start to appear in the baseline,and the phases of several peaks are not perfect.
in trace (d) of Figure 3.1. However, there is still some small distortions in the base-
line. The reason is that even though we eliminate the numerical instability introduced
by Eq. 3.2, instability due to the ill-conditioning of large linear phase corrections for
noisy signals with broad peaks is still present. FDM averaging procedures can be
used if one wish to further suppress the instabilities.
A last comment on linear phase correction via FDM and RRT is that the multi-
177
scale Fourier basis should not be used. Even though it can improve the accuracy
of the parameters for the narrow features, this is achieved by using broad poles to
describe the background and features outside the spectral window in lower resolution.
These broad poles are often some effective representation of more complicated fine
structure of actual features. Linear phase correction of these “unphysical” poles is
meaningless and likely to introduce severe baseline distortions.
3.3 Nontrivial Projections by FDM
In addition to the trivial projections such as the conventional 1D projections
(Eq. 2.67), FDM can be also used to construct some non-trivial projections. One of the
most useful projections for NMR is the 45o projection for J-experiment [72, 71, 73, 90].
It was demonstrated in that highly resolved absorption-mode 45o projection can be
obtained by FDM, while for FT construction of absorption-mode 45o is impossible and
only skew 45o projection of an absolute value 2D-J spectrum can be obtained [118]. In
this section, we will first outline the mathematical approach for constructing a general
nontrivial projection along an arbitrary time vector, then focus on its application to
obtain highly resolved homonuclear correction spectra using a few TOCSY data set.
Numerical difficulties related to constructing stable nontrivial projections will also be
addressed.
Let ~p = (p1, . . . , pD) be a general N-dimensional time vector along which we want
to construct a spectral projection. pl can be any real or even complex numbers (such
as in 2D-J DOSY [96]). For example, in 2D case, ~p = (1, 0) and (0, 1) correspond to
178
the normal 1D projections along the first and second time dimensions; ~p = (1,−1) cor-
responds to the non-trivial 45o projection. Given an N-dimensional line list dk, ~ωk,
the ~p-projection of the frequencies is simply defined as,
ω~pk = ~p ~ωk =D∑
l=1
pl ωlk . (3.8)
The complex spectral ~p-projection is then3,
I~p(ω) = i∑
k
dk
ω − ω~pk. (3.9)
An absorption-mode projection can then be obtained by taking the real part of the
complex spectrum. Therefore, if we can obtain a parametric representation of the
signal, it is trivial to compute any trivial or non-trivial projections. In practice, this
procedure has difficulties due to the lack of a reliable N-dimensional FDM line list.
Instead, it is less demanding to first construct the ~p-projection Hamiltonian [73, 91],
Ω~p = ~p ~Ω =D∑
l=1
pl Ωl (3.10)
and then solve the eigenvalue problem,
Ω~p |Υ~pk) = ω~pk |Υ~pk) , (3.11)
to obtain the ~p-projection of the frequencies. The corresponding resolvent for esti-
mating complex spectral ~p-projection is then,
I~p(ω) = i
(
Φ0
∣
∣
∣
∣
∣
1
ω − Ω~pk
∣
∣
∣
∣
∣
Φ0
)
= i∑
k
(Φ0|Υ~pk)2
ω − ω~pk= i
∑
k
d~pk
ω − ω~pk. (3.12)
3Note we use the integral Fourier spectral representation instead of the discrete one as the latteris unlikely to be advantageous for non-trivial projections
179
Note that for certain types of projections such as the 45o projection of J-experiments,
ω~pk has virtually zero linewidth. Some smoothing should be used to improve the
appearance of the spectra (see Eq. 2.28).
Evaluated in Fourier basis, the operator equations of Eqs. 3.10-3.12 become nu-
merical expressions for computing general ~p-projections. Given the matrix repre-
sentations Ul for l = 0, ..., D in the Fourier basis, |ϕj), a numerical procedure of
computing a general ~p-projection is given as following,
(i) Solve D number of 1D generalized eigenvalue problems (Eq. 2.62) independently
to obtain the eigenvalues ulk ≡ e−iτlωlk and eigenvectors Blk.
(ii) Use ωlk and Blk to construct a matrix representation of Ω~p in the basis of |ϕj),
Ω~p =D∑
l=1
pl Ωl =D∑
l=1
pl
∑
k
ωlkU0BlkBT
lkU0, (3.13)
where Ωl is the corresponding matrix representation of Ωl in the Fourier basis.
(iii) Solve another generalized eigenvalue problem
Ω~pB~pk = ω~pkU0B~pk ; BT
~pkU0B~pk = 1 . (3.14)
(iv) Compute the frequencies ω~pk, and the amplitudes d~pk from the eigenvectors B~pk
(Eq. 2.56), then use Eq. 3.12 to compute the complex spectral projection I~p(ω).
Initially, FDM averaging was used to obtain an artifacts free spectrum [73]. Here
we implement the more efficient FDM2K regularization, simply by reformulating all
the GEPs involved according to the FDM2K algorithm (Eq. 2.82). For non-zero q2,
the matrices on both sides of Eq. 2.82 are asymmetric. Thus both left and right
eigenvectors need to be computed when diagonalizing Ul, especially for large regu-
180
larizations:
U†
0UlRlk = ulk
(
U†
0U0 + q2)
Rlk , (3.15)
LTlkU
†
0Ul = ulkLTlk
(
U†
0U0 + q2)
, (3.16)
where Rlk and Llk denote the right and left eigenvectors, normalized according to
LTlkU0Rlk′ = δk,k′ .
Eq. 3.13 then becomes,
Ω~p =D∑
l=1
pl
∑
k
ωlkU0RlkLTlkU0 . (3.17)
3.3.1 45o-Projections of J-Experiments.
One of the most interesting non-trivial projects for NMR spectroscopy is the
45o projections of various J-experiments. J-experiment was first proposed by Ernst
and coworkers to obtain broad-band proton-decoupled proton NMR spectrum [118].
It is hampered by the poor resolution of phase-twisted line shape, because in J-
experiments only a single purely phase-modulated data set can be acquired. Con-
ventional FT spectral analysis requires many J increments and can only provide
absolute value mode 45o-projections, which are unimpressive in resolution. The J-
spectroscopy has thus not been widely used. However, combined with FDM data
processing, J-experiments can be very useful in simplifying complicated spectra with
heavily overlapped multiplets, by first dispersing the multiplets into an additional
J-dimension and then reducing them into sharp singlets in the 45o projection.
181
-400-300-200-1000100200300
-2000
-1500
-1000
-500
0
500
1000
13
1
C / Hz
H / Hz
Figure 3.3: The singlet-HSQC spectrum of progesterone obtained using a HSQC-Jpulse sequence given in Ref. [90]. All the proton multiplets are collapsed into singlets!The 3D purely phase modulated signal consists of N1×N2×N3 = 600×64×2 pointsin the proton, carbon-13, and proton J dimensions respectively. FDM signal-lengthaveraging was used to obtain an artifact free spectrum.
FDM has been successfully applied to various J-experiments including 2D-J [72,
71] and 3D HSQC-J [73, 90]. In 2D-J experiments, ~p is set to (1,−1) to compute the
45o projection. Highly resolved absorption mode homonuclear decoupled 1D proton
NMR spectrum was obtained using as few as four J-increments [72]. In addition,
for each singlet, an estimate of the corresponding multiplet can be computed. In 3D
HSQC-J experiments, first two dimensions are proton and carbon chemical shifts and
the third dimension is the proton homonuclear J dimension. ~p = (1, 0,−1) corre-
sponds to the proton-decoupled 45o projection. Correlated with the carbon chemical
shifts, we can obtain a so called singlet-HSQC 2D double-absorption spectrum where
the proton multiplets of the conventional HSQC spectrum are collapsed into singlets
182
at the proton chemical shifts,
A(ω~p, ω2) =∑
k,k′
Re[Dk,k′]Re[
1
1 − eiτ2(ω2−ω2k)− 1
2
]
Re
[
i
ω~p − ω~pk′
]
, (3.18)
with cross amplitudes Dk,k′ computed as Eq. 2.66. It was demonstrated that the J-
dimensions did not have to be fully resolved in order to obtain a good 45o-projection.
As few as one additional increment in each J-dimension might be sufficient in many
cases. An example of singlet-HSQC spectrum is shown in Figure 3.3.
3.3.2 Singlet-TOCSY Spectra via FDM
Another interesting application of nontrivial FDM projections is the singlet-TOCSY
experiment, where two J-dimensions, combined with FDM data processing, are used
to decouple the homonuclear coupling in both proton dimensions. The closely spaced
rectangular multiplet of conventional TOCSY is reduced to a sharp and intense singlet
at the chemical shifts of two correlated protons, leading to a substantial resolution
improvement. The simplified 2D spectra are thus called singlet-TOCSY spectra.
A J-TOCSY-J pulse sequence, developed by De Angelis [119], is given in Figure 3.4
in comparison with that of the conventional TOCSY experiment. The double spin
echoes preceding each of the traditional time variable, t1 and t2, add two homonuclear
J-dimensions, t3 and t4, leading to 4D time signals. Detailed descriptions of the NMR
experimental parameters can be found in Refs. [90, 119]. Surprisingly, as few as one
additional increment along each J-dimension is sufficient. Finally, only three TOCSY
data sets, with (t3, t4) = (0, 0), (τ3, 0) and (0, τ4), are required to obtain the 2D singlet-
TOCSY spectrum.
183
n n
t2
H1
DIPSI-23 t2t1
t24t
24t
23
G
SLy xx xx
1
SLy
1
2
2
PFG
n n
H1
DIPSI-2t2t1
G
SLy xx xx
2
SLy
1
PFG
(a)
(b)
Figure 3.4: Timing diagrams of the conventional TOCSY, (a), and J-TOCSY-J ex-periments (b). The black solid icons represent 90o pulses. Scored icons representconstant amplitude FM 180o pulses. A spin lock and short Tr-ROESY purgingpulse [120] surround the isotropic mixing sequence DIPSI-2 [121, 122]. The 4-stepphase cycling φ1 = (x, y,−x,−y); φ2 = 4(x); receiver= (x, y,−x,−y) is combinedwith CYCLOPS [123] for a total of 16 scans per increment.
A 2D double 45o projection of the 4D time signal can be constructed by setting
~p1 = (1, 0,−1, 0) and ~p2 = (0, 1, 0,−1) and computing the double-absorption mode
2D spectral estimation as,
A(ω~p1, ω~p2) =∑
k,k′
Re[Dk,k′]Re
[
i
ω~p1 − ω~p1k
]
Re
[
i
ω~p2 − ω~p2k′
]
. (3.19)
Note that 2D spectral estimation with a single homonuclear decoupled proton dimen-
sion can also be obtained simply by replacing either ~p1 or ~p2 with the conventional
184
projection vectors. For example, setting ~p1 = (1, 0, 0, 0) and ~p2 = (0, 1, 0,−1) leads
to a 2D spectrum homonuclear decoupled only in the second proton dimension.
Figure 3.5 demonstrates the singlet-TOCSY using a simple spin system, 2,3-
dibromopropionic acid, with three weakly coupled protons. The pulse sequence
shown in Figure 3.4 was used with a 40.0µs spin lock, 35.0µs Tr-ROESY purging
sequence, and 53.0µs DIPSI-2 mixing time. The spectral widths are 25Hz for both
J-dimensions, and 700Hz for both proton dimensions. For each (t3, t4) increment,
200 × 80 complex data points were acquired. The regularizations were optimized for
all FDM calculations. Even though one additional increment in each J-dimension is
not sufficient to fully resolve the multiplets in J-dimension, clean and sharp singlets
can be obtained in the 45o-projections. Figure 3.6 compares the corresponding verti-
cal cross sections of the central part of the central triplet in the conventional TOCSY
and singlet-TOCSY spectra. The multiplets are not fully resolved in the indirect
dimension (t1), shown in the top trace, while nice and intense singlets are obtained
in the singlet-TOCSY, shown in the bottom trace.
The simple example of Figure 3.5 demonstrates the possibility of using FDM to
obtain homonuclear decoupled TOCSY spectra using small J-TOCSY-J 4D data sets.
However, its performance on more complex systems still needs further investigation.
There are several difficulties associated with computing 2D singlet-TOCSY spectra.
First, TOCSY spectra contain a lot of multiplets that are degenerate in both dimen-
sions, are unfavorable for FDM. Second, the scaling step of Eq. 3.13 is sensitive to
any numerical errors in the eigenvalues and eigenvectors. It is not clear how regu-
larization will effect the accuracy of projection operators as constructed. Third, it
185
3.63.73.83.94.04.14.24.34.44.54.6
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
F
1 (
pp
m)
F
1 (
pp
m)
3.63.73.83.94.04.14.24.34.44.54.6
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
F2 (ppm)
3.63.73.83.94.04.14.24.34.44.54.6
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
F2 (ppm)
3.63.73.83.94.04.14.24.34.44.54.6
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
a) b)
c) d)
Figure 3.5: A demonstration of successive simplification of TOCSY spectra usinga weakly coupled three-spin system. The 4D J-TOCSY-J signal consists of three2D TOCSY data sets, each of which consists of 200 × 80 complex data points.Panel (a) through (d) show the conventional, homonuclear decoupling in F1 only,homonuclear decoupling in F2 only, and homonuclear decoupling in both F1 and F2TOCSY spectra. The projection vectors used are: (a) ~p1 = (1, 0, 0, 0), ~p2 = (0, 1, 0, 0);(b) ~p1 = (1, 0, 0, 0), ~p2 = (0, 1, 0,−1); (c) ~p1 = (1, 0,−1, 0), ~p2 = (0, 1, 0, 0); (d)~p1 = (1, 0,−1, 0), ~p2 = (0, 1, 0,−1).
186
4.7 4.6 4.5 4.4 4.3 4.2 4.1 4.0 3.9 3.8 3.7 3.6 F1 / ppm
4.7 4.6 4.5 4.4 4.3 4.2 4.1 4.0 3.9 3.8 3.7 3.6 F1 / ppm
Figure 3.6: Vertical traces along the center peak of the central triplet of conventionalTOCSY (panel (a) of Figure 3.5) and singlet-TOCSY spectra (panel (d) of Figure 3.5).
is not clear whether and how we should regularize the second eigenvalue problem
of Eq. 3.14. Until we have solutions for overcoming these difficulties, application of
singlet-TOCSY to simplify complex systems with many heavily overlapped multiplets
will remain problematic.
187
3.4 Constant-Time NMR Signals Processed by FDM
In conventional 2D NMR experiments, two data sets, either N-/P-type phase
modulated signals or cosine/sine amplitude modulated signals, are acquired4. This
is an absolute must for FT to obtain a double absorption spectrum. However, FDM
only requires a single purely phase modulated signal, either N- or P-type, to compute
a double absorption mode 2D spectral estimation. It appears that processing N-
and P-type independent data sets, and then combining the results does not greatly
improve the FDM results. In addition, FDM might fail due to insufficient signal size
of a single phase-modulated signal (either N- or P-type). Instead, a single, twice as
long N-type (or P-type) data set would be far superior for FDM data processing,
while such as a data set will be useless for FT data processing, because only absolute
value spectra can be obtained. Therefore, it seems that the optimal data formats
for FT and FDM are different and incompatible with each other. Fortunately, for
the special case of Constant-Time (CT) NMR experiments, there exists a doubling
scheme for FDM to co-process both N- and P-type signals together as a single signal,
but twice as large. With the doubling scheme, optimal data formats for FDM and
FT become compatible. In addition, there are some special properties of CT signals,
making them particularly suitable for the FDM spectral analysis.
In this section, we will first briefly review relevant properties of CT NMR experi-
4These two types of data sets can be converted into each other simply by taking linear combi-nations. For example, given cosine and sine modulated time signal, Sc(t1, t2) and Ss(t1, t2), purelyphase modulated N- and P-type signals can be obtained as,
SP (t1, t2) = Sc(t1, t2) + iSs(t1, t2) , SN (t1, t2) = Sc(t1, t2) − iSs(t1, t2) .
188
ments, and then describe the FDM doubling scheme for processing CT signals. Both
numerical and experimental 2D and 3D examples will be given to demonstrate the
efficiency of the doubling scheme.
3.4.1 Special Properties of CT NMR Signals
Many protein backbone NMR experiments on uniformly-labelled molecules em-
ploy a fixed, or constant-time evolution period [124, 125], 2T , to encode chemical
shift information. In some cases, the constant-time period is used to decouple the
homonuclear carbon-carbon couplings, leading to sharp and intense singlets instead
of broader multiplet, as demonstrated in the CT-HSQC experiment [126]. In other
cases, constant-time evolution is encoded within a fixed delay that is already present
for coherence transfer between spins, for example, in the CT-HNCO experiment [127],
thereby minimizing the total time between excitation and acquisition.
The CT signals almost show no decay in the CT dimension, with the residual
linewidth purely determined by the quality of the magnetic field homogeneity, which
is typically excellent with modern shimming. In FT spectral analysis, the linewidth
in the CT dimension is governed by the FT time-frequency uncertainty principle. The
maximum signal length is limited by half of the total constant period, the obtainable
FT resolution is thus limited by 1/T . Apodization of the CT signal, which is to
reduce the sinc-like truncation artifacts, further broadens the lines. Therefore, longer
constant-time period is preferred from the resolution point of view, if FT is to be
used. However, there are other factors that prohibit the use of very long constant-
189
0.0
Rela
tive In
tensity
T2 = 50 ms
T2 = 20 ms
2T [ms]
1/2JCC
4010 20 30 50 60
-0.5
-0.25
0.25
0.5
0.75
1.0
Figure 3.7: The sensitivity of CT experiments as a function of the constant-timeperiod for Cα with a 35 Hz one-bond coupling to a single Cβ.
time period. The observed signal intensity decreases exponentially with respect the
constant-time period, S(2T ) ∼ e−2T/T2 , with T2 being the traverse relaxation time
constant. Thus to from the sensitivity point of view, a short constant-time period is
preferred. Accordingly, when FT is used, one has to compromise between resolution
and sensitivity and this is one of the major limitations of CT experiments.
The actual observed intensity of the aliphatic carbon is also modulated by the
one-bond coupling between Cα and Cβ, Jcc ∼ 35Hz, further limiting the usable
constant-time period for carbon:
S(2T ) = e−2T/T2 cos(2πJccT ) . (3.20)
Figure 3.7 plots the relative intensity of Cα as a function of the constant-time period
for two different relaxation time constants. To obtain a sufficient FT resolution
without losing most of the signal, 2T is typically chosen to be either 26 − 28ms or
52 − 56ms, avoiding to the nulls at 1/Jcc and 2/Jcc.
190
FDM is a true multi-dimensional method in which the obtainable resolution in
all dimensions is determined together by the total information content of the data
set. It is not necessary for the signal to be long in all dimensions in order to obtain
high spectral resolution in all dimensions. In particular, it is possible for FDM to
use the information encoded in a long dimension to enhance the resolution in the
short dimensions. Thus, combined with FDM data processing, one does not have to
compromise between sensitivity and resolution in CT experiments. Short constant-
time periods can be used to improve the sensitivity without sacrificing the resolution
in a CT dimension. An additional bonus of using FDM data processing is that the
better sensitivity, as a result of using shorter constant-time period, can be effectively
translated into a better resolution in the FDM spectra.
Last, but not the least, CT signals have two special properties that make them
particularly suitable for FDM data processing. First, CT signals have virtually zero
linewidth and thus perfect Lorentzian lineshape in the CT dimensions. Therefore,
CT signals can be handled by FDM very efficiently. Second, there is a time reverse
symmetry between N- and P-type CT data sets. It is possible for FDM to process
them together as a single data set, but twice as large. Due to the sudden convergence
property of nonlinear methods, doubling the signal size can often lead to a dramatic
improvement in the obtainable resolution. This doubling scheme will be first described
in details and then demonstrated using both numerical and experimental signals.
191
3.4.2 A Doubling Scheme for Processing CT NMR Signals
Assuming zero phase at t1 = n1τ1 = 0, t2 = n2τ2 = 0, which is readily achievable
experimentally, the N- and P-type signals for a single peak may be written as,
SN(n1τ1, n2τ2) = exp[i(ω2n2τ2 − ω1n1τ1)] , (3.21)
SP (n1τ1, n2τ2) = exp[i(ω2n2τ2 + ω1n1τ1)] , n1τ1 = 0, τ1, . . . , T ,
where t1 corresponds to the CT dimension, and t2 the acquisition time dimension.
ω2 is complex, but ω1 is essentially real5, as the imaginary part vanishes in the CT
dimension. In a conventional picture, the N-type signal appears to evolve forward
in time but with “negative frequency”. Alternatively, we can consider it to evolve
with the same “positive frequency” as P-type signal, but with negative time step:
n1(−τ1) = 0,−τ1, . . . ,−T . Therefore, by preceding the P-type data set, in the time
domain, with the N-type data in reverse chronological order, a pseudo-signal that
evolves effectively almost twice as long in CT dimension results:
SNP (n1τ1, n2τ2) = exp[i(ω2n2τ2 + ω1n1τ1)] , n1τ1 = −T, . . . , T . (3.22)
Note that this is achievable only if there is no decay in the CT dimension. Otherwise,
SNP will have a Golden Bridge shape and is inconsistent before and after t1 = 0.
Even though the combined signal SNP is twice as long, due to the time symmetry
at t1 = 0, the FT resolution in the CT dimension is still determined by 1/T instead of
1/2T . FT of such the combined signal automatically produces absorption lineshape
in the t1 dimension, which is not surprising since the purpose of acquiring both N-
5No assumption that ω1 should be real is used in FDM when CT signals are analyzed.
192
1
2
1
22
Re[I(ω)] Im[I(ω)]
0
0 0
1
2
0
1
+T
SP = e+i ω t1
SNP = e+i ω t1
-T +T
FT
FT
+T
SN = e-i ω t1
ω
ω
1
2
0
1
2
0
-ω -ω
ω
ω
FT
Re[c(t)]
Figure 3.8: Time symmetry between N- and P-type CT signals and Fourier Trans-forms of them. It shows that due to the time symmetry between N- and P-type CTsignals, FT of the combined signal, SNP , automatically gives an absorption spectrumwith vanishing imaginary part. The peak is twice as intense but the linewidth is thesame as those in absorption mode FT spectra of N- or P-type signals (indicated bythe double arrows.)
and P-type signals is to obtain an absorption lineshape in the indirect dimension.
Interestingly, this property was actually utilized, long ago, to obtain absorption-
mode lineshape in 2D NMR experiments [124, 128]. Figure 3.8 illustrates the time
reverse symmetry between N- and P-type signals and its effects on the lineshape and
resolution of the FT of the combined signal.
However, this time symmetry is totally irrelevant for FDM data processing. FDM
is a nonlinear method that is capable of producing double absorption type spectra
from a single purely phase-modulated signal. The signals effectively evolve twice as
long and thus the combined signal contains roughly twice as much information that
193
can be utilized by FDM. Due to the “sudden convergence” property of any nonlinear
method, doubling the signal size can lead to a substantial resolution enhancement.
Our procedure is clearly reminiscent of the method called “mirror-image” linear
prediction [129, 127, 130]. It has been noticed [127] that, as the CT signal does not
decay, twice as many points can be used to extrapolated the signal by LP if the data
is “reflected” in F1. Thus twice as many LP coefficients can be computed and then
used to extrapolate the data, usually be a factor of two. Once the longer signal is
obtained, the “mirror-image” part is discarded and spectrum is obtained by taking
the FT of the new extended data set. This procedure can improve the resolution in
the F1 dimension, although the final spectral resolution can not exceed that expected
from a hypothetical experiment in which data over [0,2T] is collected and transformed.
There are further limitations. Reflecting the signal also symmetrizes the noise, which
could introduce some avoidable bias.e An improved mirror-image LP will use SNP ,
to compute the LP coefficients and then use them to both forward and backward
extrapolate the signal by a factor of two to obtain an extended signal , available from
t1 = −2T, . . . , 2T , from which extended N-type and P-type signals can be obtained
by simply keeping the negative or positive halves. A more severe limitation of the
mirror-image LP procedure, which is shared by the Fourier transform, lies in the fact
that it is essentially a 1D procedure and unable to utilize the important information
that evolutions in two dimensions are correlated.
In order to process the signal SNP , only minor changes to the previous FDM2K
algorithm are required. Let’s redefine the 2D discrete time signal c(~n) = c(n1τ1, n2τ2)
to be available from n1 = −(N1 − 1), . . . , 0, . . . , N1 − 1 and n2 = 0, . . . , N2 − 1. The
194
new Fourier basis then reads,
∣
∣
∣ΨNPj
)
=M1−1∑
n1=−(M1−1)
M2−1∑
n2=0
ein1τ1ϕ1jein2τ2ϕ2j |Φn1,n2) (3.23)
= e−i(M1−1)ϕ1j
2(M1−1)∑
n1=0
M2−1∑
n2=0
ein1τ1ϕ1jein2τ2ϕ2j |Φn1−(M1−1), n2)
= e−i(M1−1)ϕ1j |Ψj(M1 − 1)) , (3.24)
where M1 ≡ N1/2, M2 ≡ N2/2, and |Ψj(M1−1)) denotes the normal 2D Fourier basis
defined in Eq. 2.50 but with shifted Krylov basis |Φn1−(M1−1), n2). j = 1, 2, ..., Kwin.
The U matrices in the new Fourier basis can be then computed as,
[
U(~p)]
jj′=
(
ΨNPj
∣
∣
∣ U (~p)∣
∣
∣ΨNPj′
)
= e−i(M1−1)(ϕ1j+ϕ1j′ ) (Ψj(M1 − 1)| U (~p) |Ψj′(M1 − 1)) , (3.25)
where the (Ψj(M1 − 1)| U (~p) |Ψj′(M1 − 1)) part can be computed efficiently using the
formulas derived previously (Eqs. 2.52-2.54), but with shifted data matrices, c(~n′ +
~p) = c(n1 −2(M1 −1)+p1, n2 +p2). e−i(M1−1)(ϕ1j+ϕ1j′ ) is simply a “phase factor” that
can be included after the (Ψj(M1 − 1)| U (~p) |Ψj′(M1 − 1)) term is computed.
Once the U matrices are available, one can solve the generalized eigenvalue prob-
lems of Eq. 2.62 and then use the Green’s Function formulas, such as Eq. 2.68, to
construct various 2D spectral estimations. The last minor modification is to redefine
the FT column vector C (Eq. 2.57) accordingly,
[
C]
j=
M1−1∑
n1=−(M1−1)
M2−1∑
n2=0
ei~n~ϕjc(~n) , j = 1, 2, ..., Kwin . (3.26)
195
0
500
250
-250
300 100
0
500
250
-250
300 100 / HzF2
/ H
zF
1
/ HzF2
/ H
zF
1
Figure 3.9: A comparison between 2D Lorentzian (left panel) and Gaussian lineshape(right panel), using a selected region of a 2D 13C Constant-Time HSQC of uniformlylabelled human ubiquitin.
3.4.3 Lineshape Modification
Although the output of FDM is expressed in terms of Lorentzians lines, it is pos-
sible to convert them into any other lineshapes such as a Gaussian lineshape. The
most conservative way is to employ the standard Lorentzian to Gaussian deconvolu-
tion [131]. A rather more aggressive procedure, and the one we have employed here,
is simply to replace each Lorentzian with a Gaussian of the same full-width-at-half-
magnitude (FWHM) and equal integral,
IG(ω1, ω2) =4
π
∑
k,k′
Re [dk,k′]
γ1kγ2k′
e−(ω1−ν1k)2/γ2lk e−(ω2−ν2k′)
2/γ22k′ , (3.27)
where νlk = Re[ωlk] and γlk = −Im[ωlk] are the line positions and linewidths computed
by FDM. The well-known star-shaped contours of the 2D Lorentzian peaks become
elliptical contours that are more easily discerned, as demonstrated in Figure 3.9.
This line shape modification is purely cosmetic in the sense that the information
on the peaks is already entirely encapsulated by the parametric description of the
196
spectrum. The downside of this more aggressive procedure is that when broad lines
interfere by overlapping Lorentzian tails, the sharper cutoff of the Gaussian may
introduce ”new” features that are misleading. Practically, well-resolved spectra are
readily obtainable by FDM and additional artifact due to Gaussian formula is often
at minimum. When the method fails to resolve all the peaks, instability might occur
locally in the unresolved region. This will be further discussed later on in this section
when we demonstrate this doubling scheme using a experimental NMR signal and
study its sensitivity to the noise level.
3.4.4 Numerical Experiments
Numerical studies on model signals plus noise have the advantage that the cor-
rect answer is known exactly. As long as this information is not employed in any
way by the algorithm, model data can be used as a powerful tool for studying and
demonstrating new computation methods. We have made an attempt to objectify the
performance of 2D FDM2K by examining model N- and P-type signals from a hypo-
thetical constant time experiment, including various amounts of uncorrelated random
noise. Figure 3.10 shows a model spectrum meant to mimic a section of a CT-HSQC
spectrum. There are 13 peaks of identical integral, with similar line widths in F2
in the range of 2.2 ± 0.2 Hz (at random) and zero line width in F1 (mimicking CT
dimension). The spectral width is 1 KHz in both dimensions. The SNR is extremely
high, around 1000:1 in the time domain. The exact spectrum, panel (a), has been
line broadened in F1 to roughly the mean width in F2 to allow meaningful contours
197
to be drawn. The absorption mode DFT spectrum is shown in panel (b), computed
from a pair of N- and P-type data with size 10× 10. Cosine apodization was used to
limit the sinc-function wiggles and zero-filling to 512× 512 points was used to obtain
same digital resolution as the FDM spectra. Some of the peaks are resolved, but
others are too close to distinguish. In panel (c) the FDM double-absorption spec-
trum from the 10 × 10 time-domain P-type data set is shown. Many of the peaks
are obtained quantitatively, but local crowding results in several pairs of peaks be-
ing fit as a single broader peak. Note in addition that the apparent width of the
single feature is less than the separation of the two genuine peaks. Convoluting this
spectrum with the transform-limited line shape still gives a good match to the actual
DFT spectrum (data not shown), but this is clearly not sufficient to guarantee that
all the information has been obtained. Finally, in panel (d) the effect of processing
a single ”NP” data set, SNP , is shown. The larger basis size (and possibly slightly
better overall SNR) allows all the peaks to be obtained correctly. With experimental
data, of course, one never knows in advance what the correct result is supposed to be!
However, by doing several FDM2K calculations with different basis size, it is possible
to identify stable recurring features, and also features that depend on the size of the
basis. As long as a feature is stable with respect to basis size and the regularization
parameter, our experience has been that it is quite accurate and reliable.
Figure 3.11 is an attempt to be somewhat more realistic. The same features are
present as in Figure 3.10, but now the noise level is substantially higher, giving around
20:1 SNR in the time domain. A DFT of a pair of 18×18 time-domain data sets shows
resolution of most, but not all, of the peaks in the absorption-mode spectrum, panel
198
c) FDMNsig=10 x10
-800 -400 0 400 800
d) FDM, Doubling ShemeNsig = 10 x10
-800
-400
400
800
-800 -400 0 400 800
a) Exact
-800 -400 0 400 800
b) DFTNsig=10 x10
-800 -400 0 400 800
-800
-400
0
400
800
F2 / Hz F2 / Hz
F1
/ Hz
F1
/ Hz
Figure 3.10: Demonstration of FDM doubling scheme for analyzing CT NMR signals,using a model signal with a high SNR of 1000:1 in the time domain (first point, seetext for details). In all FDM spectra, the linewidth in F1 (CT dimension) is smoothedto the average linewidth of 2Hz in F2. With Nsig = 10× 10, while neither DFT nornormal FDM is able to fully resolve the most crowded region, FDM with doublingscheme is able to efficiently use the information from both N- and P-type data setstogether and provide a fully resolved spectral estimation, shown in panel (d). Notethat 2D DFT of the combined signal SNP will be identical to the DFT spectrumshown in (b), given that the factor of 2 in intensity is correctly taken into account(data not shown).
199
(a). The FDM2K calculations all employ the doubling scheme, but now the results
are less impressive. Using only a 10 × 10 signal there is now inadequate information
for FDM2K to accurately identify all the peaks, as shown in panel (b). Increasing
the signal size to 14 × 14 in panel (c) sharpens up most of the features. But only
when using the same 18 × 18 data set as used in the DFT calculation again allows
all the features to be identified, as shown in panel (d), although there are still some
errors in the widths and intensities of the most crowded peaks. Further increasing
the signal size only results in minor improvements and is evident that the calculation
has converged to the true features.
Another interesting observation of the numerical experiments of Figure 3.11 is the
different behaviours of peaks in crowded regions and isolated ones, due to the local
convergence property of FDM. Isolated features tend to converge first and stay stable
with respect to the change of signal size, while crowded regions are more difficult to
converge and sensitive to the change of signal size until fully resolved. By computing
a series of FDM spectra with varying signal size, one can quite reliably decide whether
a particular feature is genuine or not. In addition, sensitivity of unconverged features
is not limited to varying signal size. Other FDM parameters such as basis size,
basis density and regularization can also be used to probe the different stabilities
between converged and unconverged features. Again, in practice when we use FDM
to analyze experimental NMR signals, it is always necessary to produce several, if
not many, FDM spectra with different signal size and computing parameters before
we want to draw some reliable conclusions on the results as obtained.
200
b) FDMNsig=10 x10
-800 -400 0 400 800 -800 -400 0 400 800
-800 -400 0 400 800 -800 -400 0 400 800
-800
-400
0
400
800
-800
-400
0
400
800
F1/ H
zF
1 / H
z
F2 / Hz F2 / Hz
c) FDMNsig=14 x14
d) FDMNsig=18 x18
a) 2D FTNsig = 18 x18
Figure 3.11: Demonstration of FDM doubling scheme for analyzing CT NMR signals,using the same model signal as Figure 3.10 but with a low SNR of 20:1 in the timedomain. Panel (a) shows a contour plot of absorption mode DFT spectrum obtainedusing 18 × 18 complex data points (both N- and P-type signals), with the contoursdrawn low enough t show some of the noise. Panel (b) through (d) show the FDMspectra obtained using successively larger data sets, from 10 × 10 up to 18 × 18.The doubling scheme was used in all calculations. A continuous improvement in bothintensity and resolution is seen with increasing signal size. With 18×18, all peaks areessentially resolved by FDM. Further increase of signal size will slightly improve thepeak lineshapes and intensities, but the resolution remains similar (not shown). Alsonote the different behaviours of the peaks in crowded regions and isolated ones, whichis an important way of checking the convergence of FDM calculations (see text).
201
3.4.5 Application to a CT-HSQC Experiment
The 2D Heteronuclear Single-Quantum Correlation (HSQC) experiment [132] is
one of most basic and most useful NMR experiments. HSQC correlates heteronuclear
resonances with proton resonances by transferring the polarization between heteronu-
clear spins, typically 1H and 13C or 15N. It is routinely used to obtain the backbone
assignments of proteins due to the high sensitivity. Together with 2D Heteronuclear
Multiple-Quantum Correlation (HMQC) [133], they are integral components of all
heteronuclear 3D and 4D experiments [11].
Constant-Time HSQC (CT-HSQC) [126, 11] is often used to simplify the spectra
by collapsing the multiplets along the carbon dimension into sharp and intense sin-
glets, making it easier to resolve and assign all the resonances. Here we demonstrate
the power of FDM with doubling scheme using a CT-HSQC experiment of a model
protein, human ubiquitin. All the 1H-13C CT-HSQC spectra shown were obtained
using a 500 MHz Varian UnityPlus spectrometer equipped with a standard 5 mm
HCN triple resonance probe. The sample was 700 ml of a 1 mM solution of 100%
doubly labeled (13C and 15N) human ubiquitin in D2O (VLI Research) at 24 oC. A
slightly modified pulse sequence, shown in Figure 3.12, was used to acquire the sig-
nals. The partly selective noco and co pulses [136], of duration 200µs, either invert
the aliphatic region of the carbon-13 spectrum and not the carbonyls, or vice-versa.
Their only role here is to provide a spectrum with zero phase correction in F1 , with
no Bloch-Siegert phase shifts [137], and no loss of signal intensity over the slightly
inhomogeneous radio-frequency field on the carbon-13 channel. Their short pulse
202
H1
PFG
C
N15
13
C’13
τ ττ τ t 2
t 1
4-
G1 G2
T t 1
4t1
4
Dec.
x y
t 1
4-T
y
φ1
φ1φ2 φ3
φ4 φ5
y
Figure 3.12: Modified CT-HSQC pulse sequence to observe correlations between Hα
and Cα spins in a protein. Aside from the minor addition of a pair of partly selectivepulses, denoted noco and co, and the broadband inversion pulses (BIPs) [134], denotedwith scored icons, the pulse sequence is standard. The 200µs noco and co pulses havebeen developed for ultra-short CT experiments, in which longer pulses reduce theavailable number of increments. They work to invert the aliphatic region of the 13Cspectrum without inverting the carbonyl region (noco) or vice-versa (co). The BIPscleanly invert the entire chemical shift range of the indicated nucleus. All these pulsesshow superb compensation for RF inhomogeneity. By using the pair of partly selective180s, all phase shifts are removed, leading to a spectrum with zero phase correctionin the indirect dimension. The proton 90o time was 10µs and the carbon-13 90o timewas 15µs. The co and noco shaped pulses are applied with a maximum of 10 kHz; theother pulses are applied at full power. All the spectra shown result from two scansper increment, in which the phase φ3 = x, y; receiver=x,−x. φ1 and the receiverphase are inverted for each t1 increment. Cosine/sine modulated data are acquiredaccording to the States-TPPI scheme [135].
width ensures maximal use of 2T period for data acquisition.
As the brief gradients during the longitudinal states of the INEPT and reverse
INEPT steps were enough to suppress the residual HDO resonance, amplitude mod-
ulated data sets were in fact acquired. The cosine and sine data sets required two
scans each to complete the simplest difference phase cycle to select the 13C-bearing
protons. These two data sets were then combined to form pseudo N- and P-type data
sets, which were then processed as above. Calculations with actual gradient-selected
203
data showed no significant difference as long as the experiments were done carefully
and the same number of increments obtained. For ultra-short CT experiments, the
loss of time during the actual pulsed field gradient is disadvantageous.
Noise Performance
The numerical experiments of the last section carry over fairly well to actual CT-
HSQC data. Figure 3.13 shows a zoomed section, in a crowded portion of the Cα
region, of a 1H-13C CT-HSQC spectrum. A signal with 1024 × 90 complex data
points (for both N- and P-type data sets) was acquired with a short constant time
of 2T = 7.5 ms, well below the null in the CT transfer function at ∼ 13 ms, and
almost a factor of four shorter than the typical time of 2T = 1/Jcc ≈ 26.4 ms. The
total experimental time is about 7 minutes. Lowering the power of the last proton
90 pulse has mimicked the effect of poorer sensitivity. The 2D FT spectra are on the
top panels and the 2D FDM2K spectra from the same data are directly underneath.
The FT spectrum from the FID of the first increment, with some blank down-field
region included for a noise reference, is shown along the top of each of the series.
As the effective read pulse is reduced to 11o and finally 5o, corresponding to a rela-
tive SNR five and ten times lower, respectively, it is clear from the first increment that
the data becomes quite noisy. Nevertheless, FDM2K with the new doubling scheme
behaves remarkably predictably, with the achievable resolution gradually diminish-
ing, and weaker features disappearing along with the noise when the regularization
is used. There is clearly no catastrophic loss of performance, and the resolution in
many regions is always preferable to the FT result. The fine structure along the
204
2D FT
2D FDM
55
63
C/p
pm
13
55
63
C/p
pm
13
4.0H/ppm15.0 4.0H/ppm15.04.0H/ppm15.0
7 6 5 4 7 6 5 4 7 6 5 4
H/ppm1H/ppm1 H/ppm1
90 read pulseo 11 read pulse
o5 read pulse
o
Figure 3.13: Demonstration of the noise performance of FDM2K. The top traces showthe FT of the FID from the first increment, with some down-field region included toshow the noise level. The contour plots below show the 2D FT (top series) and2D FDM2K (bottom series) spectra. The pulse sequence of Figure 3.12 has beenused with the power of the 10 ms read-out proton 90o pulse attenuated to achievethe desired flip angle. The 2D FT spectra are shown in the top series, and the2D FDM2K spectra, from the doubled NP data set, are shown directly underneath.Regularization has been increased as the SNR decreases from left to right. Two scansper increment were recorded for the total of 90 increments, for the sine and cosinedata sets. (a) 90o read pulse. (b) 11o read pulse (-18 dB). (c) 5o read pulse (-25 dB).
205
proton dimension, a known result of a flip angle effect [138] when the small flip angle
read pulses are used, is blurred away by the regularization. The result is a set of
nearly structureless ellipses centered on the correct chemical shifts. Particularly with
the noisier spectra, attempts to resolve the proton multiplet structure, and simulta-
neously separate the close carbon-13 shifts, have not been successful.
Ultra-short Constant Time Performance
Attenuation of Cα magnetization during the constant time period is a common
problem in larger proteins, where the T2 losses can be severe. However, using larger
B0 fields in conjunction with FDM2K may allow very short constant time periods,
greatly improving sensitivity without the need for deuteration. This is in contrast
to the usual fixed value of 2T ≈ 26.4 ms, independent of the field strength. The
difference is that by operating with (much) shorter periods than the null condition at
13 ms, shown as the shade area in Figure 3.7, the delay can be shortened as much as
we like, fixing only the number of increments obtained. For example, if a decent CT-
HSQC spectrum can be obtained using 2T = 4.25 ms at 500 MHz, then only 2.6 ms
would be required at 800 MHz. The relative basis function density versus the number
of local peaks is identical. However, there can be a great improvement in performance
both on account of the superior intrinsic sensitivity of the higher field strength, and
the reduction in relaxation losses. The local nature of the convergence of FDM means
that many of the spectral regions can be essentially completely resolved, leaving a
few dense regions that may require a longer experiment, some editing techniques, or
a higher-dimensional experiment to make headway. Alternatively, the same 4.25 ms
206
experiment may, at higher field, allow completely converged spectra to be obtained.
The ability to achieve high resolution with very short constant time periods could
establish FDM as a key component in successful NMR of larger proteins at high field.
Human ubiquitin is not a rapidly-relaxing protein, but can be studied simply to
assess the degree of separation of the resonances that can be obtained with ultra-
short constant times. Figure 3.14 panel (a) shows the conventional absorption-mode
spectrum in the Cα region of the CT-HSQC spectrum using 26.4 ms and the pulse
sequence of Figure 3.12. Two scans per increment for the N- and P-type data were
used. This is our reference spectrum for the study. Many of the 2D peaks are
clearly resolved, with a couple of denser regions. In the right-hand panel, (b), the
FT spectrum obtained with 2T = 4.25 ms is shown. Many of the distinct peaks have
coalesced because of the reduced resolution along the dimension. In panel (c) FDM
absorption mode spectrum of processing the P-type data only is shown for the same
4.25 ms data set. A similar result was obtained with the N-type data (not shown).
The spectrum shown is sensitive to changes in FDM processing parameters, and would
not be considered reliable. Many regions of the spectrum remain unresolved, as the
basis density is apparently too low to capture the local number of peaks. There are
only eight basis functions along the Cα dimension in the region displayed. Finally in
panel (d) the FDM2K result using the same data, but now with doubling scheme, is
displayed. Clearly the combined signal SNP is a case where the whole is greater than
the sum of the parts. Many of the regions display essentially identical resolution to
the reference FT spectrum, panel (a). In the most crowded region the basis function
density is still insufficient to characterize all the local 2D peaks, and so a distorted
207
4.55.5 3.5
50
55
60
65
70
a) FT 2T~26.4 ms
4.55.5 3.5
55
60
65
70
b) FT 2T~4.25 ms
4.55.5 3.5
50
55
60
65
70
1H / ppm
c) FDM2K 2T~ 4.25 ms
55
60
65
70
4.55.5 3.5
d) FDM2K 2T~ 4.25 ms
C
/ p
pm
13
P-type only Doubling Scheme
1H / ppm
C
/ p
pm
13
Figure 3.14: Ultra-short constant time spectra compared with conventional measure-ment. (a) Reference 2D FT spectrum using 2T = 26.4 ms. (b) 2D FT spectrumusing 2T = 4.25 ms. (c) 2D FDM2K spectrum using the same 4.25 ms data set as in(b), processing only the P-type data. This 2D spectrum is not stable in many regionsand would not be considered reliable. (d) Result of the FDM with doubling schemeapplied to the same 4.25 ms data set. There is a vast improvement in both resolutionand stability. Comparison of reference spectrum, panel (a), shows that peaks thatare resolved are also very accurately recovered by FDM2K.
208
spectrum with some wide features is obtained. There are ways to conclusively show
that regions like this are not converged, but no simple and reliable way to correct
the inaccurate results. For example, a true resonance should have a line width of
nearly zero along the Cα dimension, with errors that depend on the noise level and
quality of the data. The ”unconverged” features have line widths that are quite
large, and that are sensitive to the exact parameters of the FDM calculation, allowing
them to be flagged as unreliable. Another approach is to use the FDM parameters
to simulate the FT spectrum, and compare the result with the actual DFT of the
data. Ideally, only featureless noise should remain after subtraction, if a perfect fit is
obtained. In unconverged windows, a substantial residual may be used as evidence
of untrustworthy local spectral features. More details can be found in Section 2.6.
Note that there are only 50 increments for the 4.25 ms experiment, making this
an ultra-fast acquisition as well - less than four minutes. The calculation of the
displayed regions using FDM2K is completed in a matter of a few minutes on either
an AMD XP 1800+ Athlon PC running Linux, or an older 533 MHz DEC Alpha
workstation. Calculation of the entire 2D spectrum takes of the order of 2-3 hours
depending on the size of the window basis. As many regions are completely devoid
of signals, however, a more realistic estimate is around 45 minutes for the relevant
areas. With an appropriate parallel architecture, the computing time could be a few
minutes for virtually any size data set, as each window is a separate job.
209
4.55.5 3.5
50
55
60
65
70
H Chemical Shift / ppm1
C C
he
mic
al S
hift
/ p
pm
13
Mirror-Image LP 2T~ 4.25 ms
Figure 3.15: A absorption mode spectrum obtained by mirror-image LP, applied tothe same 2T = 4.25 ms ultra-short CT signal used in previous example. There isessentially no resolution enhancement compared to the 2D FT spectrum show inFigure 3.14, panel (b). This is mainly due to the fact that mirror-image LP employstrace-by-trace 1D data processing and totally misses the important information thattwo dimensions are actually correlated. The filter length was optimized by trial anderror.
210
Comparison with Mirror-Image LP
There are many factors that work together to make the result shown in Fig-
ure 3.14 possible, including the two special properties of CT signals and intrinsic
high-resolving power of FDM. In particular, it is essential to process the whole two-
dimensional together and make use of the important information that the evolutions
in two dimensions are correlated. To demonstrate the point, here we applied the more
commonly employed mirror-image LP to the same ultra-short CT data set used in
Figure 3.14. The mirror-image LP implemented in NMRPipe [139] software pack-
age was used to obtained the spectrum shown in Figure 3.15. The filter order was
optimized by trial and error. Because mirror-image LP is implememnted as 1D LP
extrapolations of the F1 interferograms, rather than a true 2D method, it is not able
to use the information that both dimensions are correlated. In this case, there are
very few increments in the indirect dimension (effectively 8 data points for the region
shown, thus a maximum of 4 poles can be used), making it impossible for LP to
efficiently model the evolution along F1.
Performance on Larger Proteins
Human ubiquitin is a relatively small and well-behaved protein that is known to
give some of the best spectra ever seen. It is not obvious how FDM will perform
on larger and less well-behaved proteins, until we try the same experimental and
data processing scheme on such kind of systems. Unfortunately, highly pure doubly-
labeled proteins are not readily accessible except for molecular biologists, making
211
extensive test of FDM currently infeasible. Here we present some preliminary results
where FDM was applied to analyze CT-HSQC experimental signals of MTH1598, a
16 KDa protein with 141 residues. MTH1598 is one of the non-membrane protein of
methanalbacterium theromautotrophicum δH, courtesy of Dr. Cheryl Arrowsmith of
the Ontario Cancer Institute. Figure 3.16 compares the conventional DFT spectrum
of a long CT-HSQC experiment with 2T = 26.4 ms, with the FT and FDM spectra of
a much shorter experiment with 2T = 7.1 ms. It clearly shows that the performance
of FDM, shown in the center panel, is just as impressive. The resolution of the FDM
spectrum, obtained using 7.1 ms constant time, is even higher than that of the FT
spectrum of a much longer experiment, shown in the top panel. There is another
interesting observation. The FDM spectrum shows a weak peak in the right bottom
corner, highlighted by a rectangular box, which is missing in the high-resolution FT
spectrum. For comparison, the bottom panel shows the FT spectrum using 7.1 ms
constant time. The peak is now present, proving that it is genuine. However, the
resolution is much worse. For example, the two peaks highlighted by the square box
are not resolved in the low-resolution FT spectrum, but well resolved in both the
FDM and high-resolution FT spectra. This is exactly one of the major limitation of
CT experiments that we discussed in the beginning of this section: if FT is used, one
has to compromise between resolution and sensitivity. Therefore, FDM might even
be more useful for analyzing larger proteins, especially those that relax very fast. In
this case, FDM data processing, combined with short or ultra-short constant time
experiments, is the optimal scheme with both good sensitivity and high resolution.
212
-2800-2600-2400-2200-2000-1800-1600
3500
4000
4500
5000
5500
-2800-2600-2400-2000-1800
4000
4500
5000
5500
3500
-2200-1600
-2600-2400-2000-1800
3500
4000
5000
5500
4500
-2200-1600 1H / Hz
13C
/ H
z1
3C
/ H
z1
3C
/ H
z
DFT: 2T = 26 ms
DFT: 2T = 7.1 ms
FDM: 2T = 7.1 ms
Figure 3.16: Comparison of CT-HSQC spectra of MTH1598 obtained with 26.4 msconstant time and processed by FT (top), with those obtained with 7.1 ms constanttime and processed by: FDM with doubling scheme (center) and 2D FT (bottom).Short constant time, combined with FDM data processing, seems to optimal and canprovide both high resolution and good sensitivity.
213
3.4.6 A Quadrupling Scheme for 3D NMR Experiments with
Two Constant-Time Dimensions
We have demonstrated that FDM, as a true two-dimensional method, can have a
substantial advantage over FT spectral analysis or any other 1D methods like mirror-
image LP. The advantages of true multidimensional spectral analysis will be even
more enormous when more dimensions are involved. In this section, we extend the
FDM doubling scheme for 2D CT signals to a quadrupling scheme for 3D NMR exper-
iments with CT evolution implemented in both indirect dimensions. The achievable
resolution enhancement will be really astonishing.
Extension of 2D FDM to the 3D case is straightforward. A third evolution operator
is introduced and the formulas derived in Section 2.3 for the 2D case can be easily
modified accordingly. Additional details can be found in Refs. [73] and [91]. For a
3D NMR signal with two CT indirect dimensions, all the four 3D data sets6 can be
combined together and processed by FDM as a single data set, but four times as
large. The doubling scheme of 2D thus becomes a quadruple scheme in 3D. Same
minor changes as those in the 2D case need to be applied to the original 3D FDM
formulas. Mathematical derivations are very similar and thus not given again here.
The example used to demonstrate the quadrupling scheme is a double constant-
time 3D HNCO experiment of human ubiquitin. HNCO is also one of the most
basic 3D heteronuclear experiments for obtaining the protein backbone chemical shift
6In 3D NMR, in order to obtain absorption lineshapes in all three dimensions in the 3D FTspectra, either cosine/sine or N-/P-type signals are acquired for both indirect dimensions, leadingto 2 × 2 = 4 3D data sets in total.
214
assignments. It correlates the chemical shifts of (15N)H, 15N and 13C(O) of previous
residue, leading to nice singlets at the chemical shifts of 1HNi -15Ni-
13C(O)i−1. HNCO
experiments have good sensitivity and are thus chosen for this demonstration. The
standard (single) CT-HNCO [127, 11] pulse sequence was slightly modified [76] by
implementing the CT evolution to encode the CO chemical shift information. The
signals were all acquired on a 500 MHz spectrometer equipped with a normal probe
at room temperature. 8 scans were acquired per increment. The sample was 1
mM uniformly labeled human ubiquitin. States-TPPI was implemented in the CO
dimension and gradient select was used in the 15N dimension. The constant-time
period was fixed at the standard value of 2T = 48 ms for the 15N dimension to
maximize the polarization transfer. A very short constant time period of 2T = 10 ms
was used in the 13C(O) dimension, to minimize the intensity lost due to the transverse
relaxation. A maximum of 6 13C(O) increments can be acquired with a spectral width
of 1 KHz. The spectral width was 2 KHz in the 15N dimension, and, in principle,
up to 48 increments could be acquired. However, only 8 increments were actually
acquired in the 15N dimension to minimize the experimental time, and also to study
the low limit of signal size required for FDM to fully resolve all the resonances for the
system under study. The spectral width in proton dimension was 10 KHz and 1024
complex points were acquired. At the end, a 3D signal of size (H, C, N) = 1024×6×8
(for each of four data sets) was acquired in about 25 minutes.
The 3D FT spectrum were obtained using cosine weighting functions along all
dimensions. The signal was zero-filled to 128 × 128 in the 13CO and 15N dimensions
to provide sufficient digital resolution for smooth contours to be drawn. The FDM
215
100 200 300 400 500 600 700 800 900
Re[C(n1,0,0)
Im[C(n1,0,0)
n1
Inte
ns
ity
(A
. U
.)
Figure 3.17: The real and imaginary parts of the first FID of the 3D HNCO signalused in Figures 3.18-3.20. Note that the signal already decays to below noise levelby about half of the full length. The tail part of the signal contains mainly noise andshould not be include in FDM calculation.
spectra were obtained using the FDM2K algorithm with the quadruple scheme. Reg-
ularization was optimized, with the final regularization level q2 = 10−4 (see Eq. 2.83).
As the signal was very short in the indirect dimensions, the 3D spectral windows
were chosen to cover the whole spectral ranges, leading to a basis size of 5 × 7 in
the (C,N) dimensions7. In the proton dimension, fully decayed signals were acquired
and thus the tail parts were mainly noise, as shown in Figure 3.17. Acquiring extra
points in the time domain helps to improve the digital resolution without sacrificing
sensitivity for FT with proper apodizations. However, the noisy tail part of the sig-
7With doubling scheme implemented in both (C,N) dimensions, a combined signal with 11 × 15(C,N) increments results. Only even number of increments can be used by FDM, Nl = 2 Ml. Themaximum basis size is thus M2 × M3 = 5 × 7.
216
nal does not contain any meaningful information and should not be included in FDM
calculations. There are several severe drawbacks for including the noisy tail in the
FDM calculations. First, it introduces more noise to the U matrices, making them
more ill-conditioned. Second, it increases the total basis size that has to be used,
making the problem even more overdetermined. Third, it increases computational
cost. Therefore, in practice, it is necessary to plot out, typically, the first FID for
choosing the optimal signal size to be used in the proton dimension. For example,
simply by looking at the plot of Figure 3.17, one can easily estimate that 500-600
data points should be using in proton dimension. When the total size the multi-
dimensional signal is sufficiently large, the FDM results are typically insensitive to
the actual number of proton increments used, as long as it is in a reasonable range.
This can be used as an indication of the convergence of FDM calculations. The 2D
projections shown in Figure 3.18-3.20 were computed using 500 increments in pro-
ton dimension. Thus the maximum basis size in proton dimension for the region
shown was Nb ∼ 500/2 × 2000Hz/SW = 50. Four overlapping windows with 20
basis functions per window were actually used. The total basis size for each window
was 20 × 5 × 7 = e700. The whole calculation took about 40 minutes on a small
PC Linux workstation with an AMD XP 1800+ CPU. Calculations were also carried
out using slightly different numbers of proton increments between 400 and 600 data
points. They produced very similar spectral estimations, showing that FDM already
converged. As there is no direct way to plot the 3D spectra, the 2D projections along
three planes, namely, (H,C), (H,N), and (C,N), are shown instead in Figures 3.18-
3.20. All the FDM projections shown are properly smoothed in the CT dimensions for
217
13
C / H
z1
3C
/ Hz
1H / Hz
-2500 -2000 -1500 -1000 -500
-400
-200
0
200
400
-2500 -2000 -1500 -1000 -500
-400
-200
0
200
400
3D FTNsig =1024x6
3D FDMNsig = 500x6
Figure 3.18: Comparison of 2D projections of the 3D FT and 3D FDM spectra in theHC plane, obtained from a short 3D double constant-time HNCO signal (see text fordetails about the experiment and data processing).
218
15N
/ Hz
15N
/ Hz
1H / Hz
-2500 -2000 -1500 -1000 -500
-2500 -2000 -1500 -1000 -500
-800
-400
0
400
800
3D FTNsig =1024x8
3D FDMNsig = 500x8
-800
-400
0
400
800
Figure 3.19: Comparison of 2D projections of the 3D FT and 3D FDM spectra in theHN plane, obtained from a short 3D double constant-time HNCO signal.
219
15N
/ Hz
-400-2000200400-1000
1000
15N
/ Hz 0
13CO / Hz
3D FTNsig = 6x8
3D FDMNsig = 6x8
-400-2000200400-1000
0
1000
Figure 3.20: Comparison of 2D projections of the 3D FT and 3D FDM spectra in theCN plane, obtained from a short 3D double constant-time HNCO signal. There areonly 6×8 points in this plane, making it the most troublesome plane for FT. However,FDM, as a true 3D method, is able to use the information from the whole 3D data setand provide good resolution in all dimensions, including these two extremely shortdimensions. The resolution enhancement is truly astonishing!
220
better plotting. The Lorentzian lineshapes were converted to the Gaussian lineshapes
according to the procedure described previously (Eq. 3.27).
Figure 3.18 compares the 2D FT and FDM projections in the (H,C) plane. We
can clearly see the effects of the FT time-frequency uncertainty principle applying to
each dimensions independently: the resolution is high in the proton dimension where
long signal exists, while the carbon dimension is poorly resolved as there are only 6
increments. The overall resolution of the FT projection is very poor, making it use-
less for obtaining chemical shift assignments. However, the FDM projection, obtained
using the same data set, shows extremely high resolution in both dimensions. All the
peaks are fully resolved and can be easily assigned. This is possible because FDM is a
multidimensional method. The resolution in all dimensions is determined together by
the total information content of the multidimensional signal. Even though there are
only 6 increments in the 13C(O) dimension, the total size of the signal is sufficiently
large for FDM to pin down all the 3D features and thus very high resolution can be
obtained in both the proton and the 13C(O) dimensions. Similar pictures are seen
in Figure 3.19, where we compare the FT and FDM projections in the (H,N) plane.
What is truly astonishing, at least to those who are used to FT spectral analysis, is
the FDM projection in the (C,N) plane, shown in Figure 3.20. In this plane, there
are only 6 × 8 increments. There is almost no usable information in the FT pro-
jection. Surprisingly, and not surprisingly, the FDM projection in (C,N) plane are
just as highly resolved as projections along the other two planes, demonstrating the
substantial advantage of true multidimensional data processing. It is predictable that
any methods that fail to utilize this advantage could not possibly obtain comparable
221
results to what has been shown here. What should also be emphasized are the two
other essential properties of FDM: solving the nonlinear fitting problem by linear al-
gebra and local spectral analysis via Fourier basis. Without these two properties, the
scale and complexity of the problem would make any methods practically infeasible.
This is one of the main reasons why methods like, Maximum Entropy Reconstruction,
even though a multi-dimension method, relying on nonlinear optimizations and at-
tempting to fit the whole signal in a single shot, could not be used to analyze realistic
data sets. It should be pointed out that 3D FDM is the first true three-dimensional
method that has ever been successfully applied to analyze experimental NMR signals
and produce meaningful results.
Validity of the FDM Results
Unfortunately, it is difficult to directly check the validity of the FDM results,
partially due to the lack of comparable references. However, there are various ways
to validate the results indirectly, as discussed in Section 2.6. We have already checked
that the results were stable with respect to varying signal size in the proton dimension,
and other FDM parameters such as basis size, basis density and regularization level.
Here we present some additional results using the checking mechanism described and
demonstrated in Section 2.6.4.
After all the spectral parameters Dk,k′,k′′, ω1k, ω2k′′, ω3k′′ are computed, in ad-
dition to estimating the infinite time DFT spectra, which have been shown in Fig-
ure 3.18-3.20, we can also estimate various finite time DFT spectra using,
222
-2400 -2200 -2000 -1800 -1600 -1400 -1200 -1000
-400
-200
0
200
400
-2400 -2200 -2000 -1800 -1600 -1400 -1200 -1000
-400
-200
0
200
400
13
C / H
z1
3C
/ Hz
1H / Hz
FDM Estimation
2D DFTNsig = 1K x 6
Figure 3.21: Comparison of the FDM estimation of the finite (H,C) FT projectionwith the actual (H,C) FT projection obtained from the original data. FDM estima-tion was computed using Eq. 3.28 with (α, β) = (1, 2); N1 = 1024, N2 = 6 and thespectral parameters Dk,k′,k′′, ω1k, ω2k′′, ω3k′′ computed from the short constant-time3D HNCO signal (see text for details). The FDM estimation agrees with the actualFT projection very well, proving that FDM does provide a reasonably good fit of the3D signal. Similar comparison was also made along (H,N) plane, which also shows avery good match (data not shown).
223
-400-2000200400-1000
-500
0
500
1000
-400-2000200400-1000
-500
0
500
1000
FDM Estimation
3D DFTNsig = 6x8
15N
/ Hz
13CO / Hz
15N
/ Hz
Figure 3.22: Comparison of the FDM estimation of the finite (C,N) FT projectionwith the actual FT projection obtained from the original data. FDM estimation wascomputed using Eq. 3.28 with (α, β) = (2, 3); N2 = 6, N3 = 8 and using the spectralparameters computed from the short constant-time 3D HNCO signal. It appearsthat the matching is reasonable but not as good as in (H,C) plane. This is becausewater signal leaks into all (C,N) planes in the FT spectra, while such a leak is totallyeliminated in FDM (see text).
224
-400-2000200400-1000
-500
0
500
1000
-400-2000200400-1000
-500
0
500
1000
FDM Estimation(using 2T = 10 ms, Nsig = 6x8)
3D DFT 2T = 24 ms, Nsig = 12x32
15N
/ Hz
13CO / Hz
15N
/ Hz
Figure 3.23: Comparison of the estimated “high-resolution” FT projection with theactual FT projection obtained from a long experimental signal with 12 × 32 incre-ments in (C,N) dimensions. FDM estimation was computed using expression Eq. 3.28with (α, β) = (2, 3); N2 = 12, N3 = 32 but using the spectral parameter that werecomputed from the short signal with only 6 × 8 (C,N) increments. The matching isstill very well, further proving that FDM does provide a very accurate and reliable fitof the 3D signal. Note that the intensities in the actual FT projection is lower dueto the relaxation during the longer CO constant time (2T = 24ms vs. 2T = 10ms).
225
ANα,Nβ(ωα, ωβ) ≈ τ1τ2τ3
∑
k,k′,k′′
Re[Dk,k′,k′′] (3.28)
Re
[
1 − eiNατα(ωα−ωαk)
1 − eiτα(ωα−ωαk)− 1
2
]
Re
[
1 − eiNβτβ(ωβ−ωβk′)
1 − eiτβ(ωβ−ωβk′)− 1
2
]
,
where (α, β) can be set to (1,2), (1,3), and (2,3) to compute the estimated finite FT
projections along (H,C), (H,N) and (C,N) planes respectively. Apodization functions
such as exponential and trigonometric functions can be also included analytically
into Eq. 3.28. All the results shown in Figures 3.21-3.23) were obtained with cosine
window functions in all dimensions, implemented either numerically for the actual
DFT calculations or analytically for computing the FDM estimations.
Figure 3.21 compares the FDM estimation of the finite FT projection with 1024×6
increments in (H,C) plane, to the actual FT projection obtained from the experimen-
tal data of the same size. The FDM estimation was computed using the spectral
parameters obtained from the short HNCO signal with the same processing parame-
ters used to produce Figures 3.18-3.20. The estimated and actual projections match
with each other very well, proving that FDM does provide a good fit of the experi-
mental signal. Similar comparison was also done in (H,N) plane, and the matching
quality was similar (data not shown). Quantitatively, the relative matching errors, as
defined in Eq. 2.111, are only 1.8% and 1.7% respectively, which are below the noise
level. Figure 3.22 compares the FDM estimation of the finite FT projection along
(C,N) plane with 6×8 increments to the actual FT projection. The mismatch is much
higher than those in the other two planes, with relative matching error being 7.5%.
However, this increase of mismatch does not mean that the parameters in (C,N) di-
226
mensions are less accurate. Instead, most of the increased mismatch is due to the
water signal that leaks to every (C,N) plane in 3D FT spectrum8, while for FDM, the
use of the Fourier basis followed by diagonalization is very efficient in suppressing the
interference of the features outside of the spectral window. To further support the
quality of FDM fit, we used the same spectral parameters, that were computed from
the short HNCO signal with 6× 8 (C,N) increments, to estimate a “high-resolution”
FT projection corresponding 12 × 32 (C,N) increments, and compared it with the
actual FT projection obtained from a much longer experiments, where the constant
time period in 13C(O) dimension was 2T = 24ms and 12× 32 (C,N) increments were
actually acquired. The results are shown in Figure 3.23. Due to the transverse re-
laxation, the intensities in the actual FT projection are lower. Otherwise, the two
projections match with each other very well, providing a strong evidence that the fit
computed by FDM is an accurate and reliable fit of the 3D signal.
3.5 Summary and Remarks
In this chapter, we applied the FDM and RRT algorithms to processing in 1D and
multidimensional NMR signals.
Long time signals are typically available in 1D NMR and high resolution FT
spectra can be obtained, leaving little room for resolution enhancement. However,
conventional FT spectral analysis can still have problems in certain cases. For ex-
ample, large linear phase corrections give rise to severe phase rolls in the baseline of
8The water signal can be actually seen in th plot of first FID as low frequency oscillations, asshown in Figure 3.17.
227
the FT spectra, because of the point-by-point fashion of linear phase correction in
FT. FDM and RRT can recognize the individual peaks and correct the linear phase
in a more consistent way. The phase roll due to point-by-point phase correction is
completely removed. For noisy signal with heavily overlapped features and strong
background, there are still some small instabilities problems in the FDM/RRT linear
phase corrected spectra, which can be further suppressed by averaging.
Multidimensional NMR signals are typically truncated in the indirect dimensions
due to practical limitations including expensive experimental time and limited long-
term instrumental stability, resulting in poor FT resolution in the corresponding fre-
quency dimensions. FDM is a true multidimensional method and can have substantial
advantages over FT in these cases. FDM is capable of analyzing the whole data set
to pin down the intrinsically multidimensional features. The obtainable resolution in
all dimensions is determined together by the total information content, or, roughly
speaking, the total size, of the signal. Multidimensional NMR signals typically have
a large total size due to the long acquisition time dimension and thus contain a lot
of information that can be used by multidimensional methods like FDM and RRT to
significantly enhance the resolution in the truncated dimensions.
Nontrivial projections can be constructed by FDM, among which the 45o pro-
jections of J-experiments are particularly useful. By constructing 45o projections,
homonuclear coupling are decoupled and intensive singlets result. FDM has been
successfully applied to various J-experiments including 2D-J, 3D HSQC-J and 4D J-
TOCSY-J. In J-TOCSY-J, double 45o projections are used to decouple the homonu-
clear coupling in both proton dimensions. So called singlet-TOCSY spectra can be
228
obtained using as few as one additional increment in each J-dimension. However,
application of the method to simplifying highly complex spectra has difficulties with
degeneracy and others, and needs more investigations.
In this chapter, we have been focusing on applying FDM to analyzing a special
type of NMR experiments that utilized Constant-Time (CT) evolutions to encode the
chemical shift information in the indirect dimensions. There are some special prop-
erties of CT signals that make them particularly suitable for FDM data processing.
Firstly, there exists an efficient doubling scheme for CT signals. A pair of N- and
P-type data sets from the same 2D CT NMR experiment are processed jointly by
FDM as a single data set, twice as large, in which the signal effectively evolves in
time for twice as long. Secondly, the signal has nearly perfect Lorentzian lineshape in
the CT dimension, thus can be efficiently handled by FDM. Applied to both model
and experimental signals, FDM with doubling scheme shows significant resolution
improvement and appears to tolerate the noise well. Very short constant time period
can be used without sacrificing the resolution, leading to higher sensitivity, especially
for proteins with short transverse relaxation time constants.
The doubling scheme can be applied to each CT dimension in higher dimensional
NMR experiments with multiple CT dimensions. When CT evolution is implemented
in all indirect dimensions, all the 2N−1 N-dimensional data sets from the same N-
dimensional experiment can be processed jointly as a single data set by FDM to
extract maximum information. This was demonstrated by a quadruple scheme for
a 3D HNCO experiment with two CT dimensions. By processing all four 3D data
sets from the same experiment together, FDM was able to provide a 3D HNCO
229
spectrum of protein human ubiquitin that are fully resolved in all dimensions, even
when there are only as few as 6 × 8 increments in the 13C(O) and 15N dimensions.
The results were validated extensively by various approaches including: (1) repeating
the FDM calculations with a wide range of different parameters such as signal size,
basis size, basis density and regularization level; (2) comparing the FDM estimations
of low-resolution and high-resolution finite FT spectra, using the spectral parameters
computed from the short signal, to the actual FT spectra obtained from both short
and long experimental signals. They all showed that the FDM spectrum already
converged and that an accurate, reliable fit of the 3D signal was obtained by FDM.
Optimization of N-dimensional NMR experiments has always presupposed that
FT will be used to obtain the N -dimensional spectrum. For example, 3D NMR is
quite often employed because decent digital resolution can be obtained without ex-
travagant experiment time. As such, a series of different 3D experiments may be used
to correlate all the desired chemical shifts, and comparing planes from experiments
taken at different times is usually mandatory to complete the assignment. Whether
this is the most profitable use of human and instrument time, once FDM is reduced
to practice, is certainly open to question. In FDM the digital resolution of individ-
ual dimensions has no direct relation to the ultimate spectral resolution. Further,
we have shown here that the best calculation is the one with the largest available
basis set that can be obtained from the measured signal. Two different 3D experi-
ments do not lend themselves to this maximal basis set in the same way that one 4D
experiment does. While extra delays may lead to some additional losses in the higher-
dimensional experiment, there is also a gain in sensitivity by having all the data in
230
one spectrum rather than divided up between several. In 3D experiments where a
set of nuclei are merely used to transfer magnetization, for example the HN(CO)CA
pulse sequence [140], the delays are already present, and removing the parentheses
to record the carbonyl chemical shift in a 4D experiment will be more sensitive and
highly advantageous, if FDM is to be used.
In closing, it is interesting to note that the time it takes to identify M random-
frequency peaks accurately in an N-dimensional experiment depends on M but not N
when FDM is employed. Indeed, as local spectral crowding is the only limiting factor
once sufficient SNR is achieved, increasing the dimensionality of the experiment, so
that the resonances are dispersed as widely as possible in frequency space, may result
in a shorter overall experiment. For backbone assignment experiments on proteins
with known primary structure, the expected value of M is known beforehand, so that
the time required to obtain a converged high-dimensional FDM spectrum should be
predictable, and scale with M. With the SNR of the first increment and the value of
M in hand, it may be possible to semi-automate the data acquisition of even quite
complex experiments. This assertion should be tested in a future series of 3D, 4D,
and 5D experiments that are specifically designed with the method in mind.
231
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